Sondre Kvamme (Uppsala University)

This is joint work with Ruben Henrard and Adam-Christiaan van Roosmalen

An important result in representation theory is the Auslander correspondence, which gives a bijection between equivalence classes of representation-finite finite-dimensional algebras \(\Lambda\) and equivalence classes of finite-dimensional algebras \(\Gamma\) satisfying \[\operatorname{gl.dim}\Gamma \leq 2 \leq \operatorname{dom.dim}\Gamma\] where \(\operatorname{gl.dim}\Gamma\) and \(\operatorname{dom.dim}\Gamma\) denotes the global dimension and dominant dimension of \(\Gamma\), respectively. Here \(\Gamma\) is characterized (up to Morita equivalence) by the equivalence of finitely presented functors \[\operatorname{mod}(\Gamma) \cong \operatorname{mod}(\operatorname{mod}\Lambda)\] where we think of \(\operatorname{mod}\Lambda\) as a ring with several objects. Unfortunately, an analogous bijection does not hold for exact categories, even if they are of finite type. For example, if \(\operatorname{mod}(\Gamma) \cong \operatorname{mod}(\operatorname{CM}(\Lambda))\) where \(\operatorname{CM}(\Lambda)\) are the maximal Cohen-Macaulay modules of a representation-finite order \(\Lambda\) with Krull-dimension \(d\), then \(\operatorname{gl.dim}\Gamma = d\), see [Iyama07] or [Enomoto18].

In our work we try to rectify this by taking into account the exact structure in the constructions.

Definition: For an exact category \(\mathcal{E}\) let \(\operatorname{mod}_{\operatorname{adm}}(\mathcal{E})\) denote the subcategory of \(\operatorname{mod}(\mathcal{E})\) consisting of all functors \(F\) admitting a projective presentation \[\mathcal{E}(-,X)\xrightarrow{\mathcal{E}(-,f)}\mathcal{E}(-,Y)\to F\to 0\] where \(f\colon X\to Y\) is admissible in \(\mathcal{E}\).

By considering \(\operatorname{mod}_{\operatorname{adm}}(\mathcal{E}) \) instead of \(\operatorname{mod}(\mathcal{E}) \) we can obtain an Auslander type result. Here the dominant dimension and global dimension is defined inside the exact category \(\operatorname{mod}_{\operatorname{adm}}(\mathcal{E}) \) in the analogous way.

Theorem A: If \(\mathcal{E}\) is an exact category with enough injectives, then \(\operatorname{mod}_{\operatorname{adm}}(\mathcal{E})\) is an exact category with enough projectives and injectives and satisfying:

1. \(\operatorname{gl.dim}(\operatorname{mod}_{\operatorname{adm}}(\mathcal{E}) )\leq 2 \leq \operatorname{dom.dim}(\operatorname{mod}_{\operatorname{adm}}(\mathcal{E}))\)

2. For any \(E\in \operatorname{mod}_{\operatorname{adm}}(\mathcal{E})\) there exists an admissible left \(\mathcal{P}\) approximation \(E\to P\) where \(\mathcal{P}\) is the subcategory of projective objects in \(\operatorname{mod}_{\operatorname{adm}}(\mathcal{E})\)

3. If \(\operatorname{Hom}_{}(E',P)=0\) for all \(P\in \mathcal{P}\), then any morphism \(E\to E'\) in \(\operatorname{mod}_{\operatorname{adm}}(\mathcal{E})\) is admissible.

Theorem B: If \(\mathcal{E'}\) is an exact category with enough projectives and injectives and satisfying \((1)-(3)\) in Theorem A, then \(\mathcal{E'}\cong \operatorname{mod}_{\operatorname{adm}}(\mathcal{E})\) for an exact category \(\mathcal{E}\) with enough injectives. Furthermore, \(\mathcal{E}\) is uniquely determined up to exact equivalence.

We also obtain a similar result without the assumption on \(\mathcal{E}\) having enough injectives, but the characterization of \(\operatorname{mod}_{\operatorname{adm}}(\mathcal{E})\) is more complicated in that case.

Using this construction, we can characterize exact structures \(\mathcal{S}\) on an idempotent complete additive category \(\mathcal{C}\) in terms of subcategories of \(\operatorname{mod} (\mathcal{C})\), inspired by similar work in [Enomoto18]. Here \(\mathcal{P}^2(\mathcal{C})\) denotes the subcategory of \(\operatorname{mod}(\mathcal{C})\) of functors of global dimension \(\leq 2\) which admit a resolution of finitely generated projectives, \(\operatorname{Tr}\colon \underline{\operatorname{mod}}(\mathcal{C})\to \underline{\operatorname{mod}}(\mathcal{C^{\operatorname{op}}})^{\operatorname{op}}\) denotes the Auslander-Bridger transpose, and for a subcategory \(\mathcal{X}\) of \(\operatorname{mod}(\mathcal{C})\) we let \(\operatorname{Tr}(\mathcal{X})\) denote the subcategory of \(\operatorname{mod}(\mathcal{C^{\operatorname{op}}})\) consisting of objects \(G\) isomorphic in \( \underline{\operatorname{mod}}(\mathcal{C^{\operatorname{op}}})\) to objects \(\operatorname{Tr}(F)\) with \(F\in \mathcal{X}\).

Theorem C: Let \(\mathcal{C}\) be an idempotent complete additive category. The association \(\mathcal{E}\mapsto \operatorname{mod}_{\operatorname{adm}}(\mathcal{E})\) gives a bijection between the following:

* Exact structures \(\mathcal{S}\) on \(\mathcal{C}\), where \(\mathcal{E}=(\mathcal{C},\mathcal{S})\) is the corresponding exact category;

* Subcategories \(\mathcal{X}\) of \(\operatorname{mod} \mathcal{C}\) satisfying the following:

1. \(\mathcal{X}\) is a resolving subcategory of \(\mathcal{P}^2(\mathcal{C})\) and \(\operatorname{Tr}(\mathcal{X})\) is a resolving subcategory of \(\mathcal{P}^2(\mathcal{C^{\operatorname{op}}})\);

2. \(\mathcal{X}\) and \(\operatorname{Tr}(\mathcal{X})\) have no objects of grade \(1\)."

*Thomas Brüstle (Bishop's University and Université de Sherbrooke)*

*Joint work with Rose-Line Baillargeon, Mikhail Gorsky and Souheila Hassoun*

First versions of exact structures as in Buchsbaum1959 and ButlerHorrocks1961 originated from studies of relative homological algebra in abelian categories. These early papers considered a mix of structures, on one hand classes of morphisms satisfying certain properties (h.f.class), on the other hand certain ('closed') subfunctors of \(\operatorname{Ext}^1\). Buchsbaum also studied a weaker notion, an f.class, which omits the condition of being closed under composition of admissible monics and epics.

The "stand alone" concept of an exact structure as a class of short exact sequences in an additive category *A* satisfying certain axioms has been laid out by Quillen, see also Keller91. The comparison to subfunctors of \(\operatorname{Ext}^1\) has been re-considered in AuslanderSolberg1993. However, the lack of a unique maximum extension-functor for arbitrary additive categories was a limiting factor in these studies until Rump2011 showed that any additive category admits a unique maximal exact structure \(\mathcal{E}_{max}\).

In BrüstleHassounLangfordRoy2018 a study of the family of all exact structures Ex(A) on an additive category A was initiated. The existence of a unique maximum exact structure allows to turn Ex(A) into a complete bounded lattice. On the side of bifunctors, this amounts to studying all closed sub-bifunctors of a unique maximum bifunctor \(\mathbb{E}_{max}\) which corresponds to the exact structure \(\mathcal{E}_{max}\). It is natural, on the bifunctor side, to extend the study to all additive sub-bifunctors, which in turn raises the question to which structure of exact sequences they correspond. We study here the corresponding systems of short exact sequences, which we call weakly exact structures, reminiscent of the f.classes studied in Buchsbaum1959. We show that:

* The weakly exact structures on A form a lattice Wex(A) , which is finite length modular when the underlying category A is additively finite.
* Wex(A) is a sub-poset of Ex(A), but it is not a *sublattice* since the construction of join is different in the two cases.
* We further study in detail the situation when A is additively finite, giving a module-theoretic characterization of the closed sub-bifunctors of \(\operatorname{Ext}^1\) among all additive sub-bifunctors.

Generalizing exact and triangulated categories, NakaokaPaly2016 introduced extriangulated categories given by an additive bifunctor \(\mathcal{E}:\) A\(^{op} \times\) A\( \to\) Ab equipped with certain extra data called a realization. We consider the equivalent notion of 1-exangulated structures as introduced in HershendLiuNakaoka2017, and by removing one of the axioms we obtain the class of weakly extriangulated structures which naturally generalize the weakly exact structures we defined earlier.

For a finite-dimensional algebra \(\Lambda\), closed sub-bifunctors of the bifunctor \(\operatorname{Ext}^1\) correspond to certain Serre subcategories of the category of finitely presented additive functors (mod \(\Lambda)^{op} \to \) Ab defined as categories of contravariant defects. This result was extended in Enomoto2018 to exact structures on additive categories. We note that the definition of contravariant defects naturally extends to the setting of weakly exact structures. Ogawa2019 defined contravariant defects in the setting of extriangulated categories, and we further extend this notion to the framework of weakly extriangulated categories. We prove that the category of defects of a weakly extriangulated structure on an additive category A is topologizing in the category coh(A) of coherent right A-modules, that is, it is closed under subquotients and finite coproducts. For coherent A-modules, we have a natural notion of subobjects and of quotients: these are defined object-wise (for objects in *A*). This allows us to define topologizing subcategories of an arbitrary (not necessarily abelian) full subcategory *C* of coh(A) as full subcategories of *C* which are topologizing in coh(A). We show that:

*Given a weakly extriangulated structure, all its substructures are uniquely characterized by their categories of defects, and each topologizing subcategory of a given category of defects defines a weakly extriangulated substructure. Weakly extriangulated substructures of a weakly exact structure are necessarily weakly exact.*

Topologizing subcategories of an abelian category form a lattice. Topologizing subcategories of the (not necessarily abelian) category of defects of a weakly extriangulated structure on *C* also form a lattice, which is an interval in the lattice of all topologizing subcategories of coh(A). Note that Serre subcategories form a subposet, but not a sublattice of this lattice. Weakly extriangulated substructures of a weakly extriangulated structure also form a natural lattice, extending the lattice of weakly exact structues. We establish lattice isomorphisms between these several lattices, summarized in the following figure:

!Screen Shot 2020-10-26 at 4.54.44 PM|690x350" My interesting research,"Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum."

Norihiro Hanihara (Nagoya University)

Calabi-Yau (CY) triangulated categories with cluster tilting objects have been an important subject in representation theory. One of the fundamental examples of such categories is the cluster category introduced in [BuanMarshReinekeReitenTodorov2006]. It is defined for an acyclic quiver \(Q\) by the orbit category \[\mathcal{D}^b(\operatorname{mod} kQ)/\tau^{-1}[1],\] which is a 2-CY triangulated category with a 2-cluster tilting object. We discuss Morita theorems for such categories which aim at characterizing cluster categories in terms of cluster tilting objects.

The only known results on such Morita theorems are the following. Let \(\mathcal{T}\) be an algebraic \(d\)-CY triangulated category with a \(d\)-cluster tilting object \(T\in\mathcal{T}\). (1) [KellerReiten2008] Suppose \(d=2\) and \(\operatorname{End}_{\mathcal{T}}(T)=kQ\) for an acyclic quiver \(Q\). Then there exists a triangle equivalence \(\mathcal{T}\simeq\mathcal{D}^b(\operatorname{mod} kQ)/\tau^{-1}[1]\). (2) [KellerMurfetVan den Bergh2011] Suppose \(d=3\) and \(\operatorname{End}_{\mathcal{T}}(T)=k\). Put \(m=\operatorname{dim}_k\operatorname{Hom}_{\mathcal{T}}(T,T[-1])\) and let \(Q_m\) be the generalized Kronecker quiver with \(m\) arrows. Then there exists a triangle equivalence \(\mathcal{T}\simeq\mathcal{D}^b(\operatorname{mod} kQ_m)/\tau^{-1/2}[1]\) for a naturally defined square root \(\tau^{1/2}\) of the AR translation.

We give a Morita-type theorem for CY categories arising from representation-*infinite* hereditary algebras and involving some roots of the AR translation, encompassing both of the above two cases.

Theorem 1. Let \(d\geq2\) and \(\mathcal{T}\) be an algebraic \(d\)-CY triangulated category with a \(d\)-cluster tilting obejct \(T\in\mathcal{T}\). Suppose \(H=\operatorname{End}_{\mathcal{T}}(T\oplus T[-1]\oplus \cdots\oplus T[-(d-2)])\) is hereditary. Then there exists a triangle equivalence
\(\)
\mathcal{T}\simeq\mathcal{D}^b(\operatorname{mod} kQ)/\tau^{-1/(d-1)}[1]
\(\)
for a naturally defined \((d-1)\)-st root of the AR translation, provided any ring summand of \(H\) is representation-infinite.

When \(d=2, 3\) this recovers the above known results when \(\mathcal{T}\) is of infinite type. We note that we still have a partial result for the Dynkin cases which gives the classification of objects of \(\mathcal{T}\).

Although the assumption that \(\operatorname{End}_{\mathcal{T}}(T\oplus \cdots\oplus T[-(d-2)])\) is hereditary looks strong, we have the following sufficient condition for this.

Theorem 2. Let \(\mathcal{T}\) be a \(d\)-CY triangulated category with a \(d\)-cluster tilting object \(T\), with \(d\geq3\). Then \(\operatorname{End}_{\mathcal{T}}(T\oplus \cdots\oplus T[-(d-2)])\) is hereditary as soon as \(\operatorname{End}_{\mathcal{T}}(T\oplus \cdots\oplus T[-(d-3)])\) is.

For example when \(d=3\), hereditaryness of \(\operatorname{End}_{\mathcal{T}}(T)\) implies that of \(\operatorname{End}_{\mathcal{T}}(T\oplus T[-1])\). Similarly when \(d=4\), semisimplicity of \(\operatorname{End}_{\mathcal{T}}(T)\) gives hereditaryness of \(\operatorname{End}_{\mathcal{T}}(T\oplus T[-1])\), thus that of \(\operatorname{End}_{\mathcal{T}}(T\oplus T[-1]\oplus T[-2])\). Consequently we have the following (1) partial common generalization of Keller-Reiten's and Keller-Murfet-Van den Bergh's established theorems, and (2) a 4-CY version of Keller-Murfet-Van den Bergh's theorem.

Corollary 3. Let \(\mathcal{T}\) be an algebraic \(d\)-CY triangulated category with a \(d\)-cluster tilting object \(T\).
1. Suppose \(d=3\) and \(\operatorname{End}_{\mathcal{T}}(T)=kQ\) for an acyclic quiver \(Q\). Then \(H=\operatorname{End}_{\mathcal{T}}(T\oplus T[-1])\) is hereditary and there exists a triangle equivalence \(\mathcal{T}\simeq\mathcal{D}^b(\operatorname{mod} H)/\tau^{-1/2}[1]\), provided \(H\) is of non-Dynkin type.

1. Suppose \(d=4\) and \(\operatorname{End}_{\mathcal{T}}(T)=k\times\cdots\times k\). Then \(H=\operatorname{End}_{\mathcal{T}}(T\oplus T[-1]\oplus T[-2])\) is hereditary and there exists a triangle equivalence \(\mathcal{T}\simeq\mathcal{D}^b(\operatorname{mod} H)/\tau^{-1/3}[1]\), provided \(H\) is of non-Dynkin type."

David Fernández (Bielefeld University)

Multiplicative quiver varieties are in a natural way Poisson varieties. Whereas this result can be obtained by performing infinite-dimensional symplectic reduction, an alternative finite-dimensional approach was given in [AlekseevKosmannMeinrenken2002] by fusing quasi-Poisson manifolds and quasi-Hamiltonian reduction. In [VandenBergh2008], M. Van den Bergh realized that these notions were shadows of their noncommutative analogues in application of the paradigm of this area, the *Kontsevich–Rosenberg principle*, whereby a structure on an associative algebra \(A\) carries algebro-geometric meaning if it induces standard algebro-geometric structures on its representation schemes. Hence, Van den Bergh defined a *double quasi-Poisson bracket* as a map \(\left\{\mkern-6mu\left\{-,-\right\}\mkern-6mu\right\}\colon A\times A\to A\otimes A\) satisfying skewsymmetry, the Leibniz rule and a suitable non-homogeneous Jacobi identity; if we require that this Jacobi identity be homogeneous, *double Poisson algebras* arise. Interestingly, both structures satisfy the Kontsevich–Rosenberg principle.

On the other hand, M. Kontsevich and Y. Vlassopoulos introduced pre-Calabi-Yau algebras because compact Calabi-Yau algebras were too restrictive for applications related to path spaces, Fukaya categories, open Calabi-Yau manifolds or Fano manifolds; in representation theory, they include proper and smooth Calabi-Yau algebras as special cases. If \(d\in\mathbb{Z}\), a *\(d\)-pre-Calabi-Yau algebra* on \(A\) is a cyclic \(A_\infty\)-algebra structure on \(A\oplus A^*[d-1]\) for the natural bilinear form \(\Gamma\) of degree \(d-1\) induced by evaluation such that \(A\) is an \(A_\infty\)-subalgebra. Alternatively, they can be defined as solutions to the Maurer–Cartan equation for the generalized necklace bracket on the space of higher cyclic Hochschild cochains. So, pre-Calabi-Yau algebras are noncommutative Poisson structures in formal geometry.

Hence, since double Poisson algebras and pre-Calabi-Yau algebras can be regarded as noncommutative Poisson structures, one should expect some relationship between them. In [IyuduKontsevich2018], N. Iyudu and Kontsevich proved there is a one-to-one correspondence between the class of non-graded double Poisson algebras and that of pre-Calabi-Yau algebras whose multiplications \(m_i\) vanish for \(i\in\mathbb{N}\setminus \{2,3\}\). This correspondence is pretty explicit: \((f\otimes g)\big(\left\{\mkern-6mu\left\{a,b\right\}\mkern-6mu\right\}\big)=\Gamma\big(m_3(b,tg,a),tf\big)\), for all \(a,b\in A\), and \(tf,tg\in A^*[[https://arxiv.org/pdf/1902.00787.pdf|-1]\). In FernándezHerscovich2019], we further extended this bijection to the differential graded setting to include \(d\)-pre-Calabi-Yau algebras, and we studied its functoriality by introducing a suitable notion of morphism of pre-Calabi-Yau algebras, based on relative Calabi-Yau structures, as introduced in [BravDyckerhoff2019] and explained at ICRA2020 Workshop by B. Keller.

In the light of these results, it is natural to ask whether there exists a link between pre-Calabi-Yau algebras and double quasi-Poisson algebras. The starting point is the fact that the homogeneous Jacobi identity of double Poisson algebras is amount to the third Stasheff identity (\(\operatorname{SI}(3)\), for short) on \(A\oplus A^*[-1]\), which is expressed only in terms of the multiplication \(m_3\) due to the constrains on the pre-Calabi-Yau algebras under consideration. Therefore, the idea is to add a suitable \(m_4\) to the pre-Calabi-Yau structure to grasp the non-homogeneous terms of the Jacobi identity characterizing double quasi-Poisson algebras. Unfortunately, this turns out to be too optimistic because such a \(m_4\) together with \(m_2,m_3\) do not satisfy higher Stasheff identities – *e.g.* \(\operatorname{SI}(7)\). In [FernándezHerscovich2020] we proved that given a double quasi-Poisson algebra \((A, \left\{\mkern-6mu\left\{-,-\right\}\mkern-6mu\right\})\), \(A\oplus A^*[-1]\), endowed with \(m_3\) as above and an infinite number of explicitly defined multiplications \(\{m_{2i}\}_{i\geq 1}\), is a (cyclic) \(A_\infty\)-algebra, thus defining a pre-Calabi-Yau structure on \(A\). Remarkably, the multiplications \(\{m_{2i}\}_{i\geq 2}\) are weighted by the Bernoulli numbers, and the key point of the proof is that the Stasheff identities hold if and only if an interesting quadratic recurrence on the Bernoulli numbers is satisfied, being closely related to the Lawrence–Sullivan differential graded Lie algebra."

Georgios Dalezios (University of Athens)

Two finite dimensional algebras \(A\) and \(B\) are said to be *singularly equivalent* if there exists a triangulated equivalence between their singularity categories, \(\mathbf{D}_{\mathrm{sg}}(A)\cong\mathbf{D}_{\mathrm{sg}}(B)\). In general, such equivalences might not preserve important properties of the algebras in question, thus it is more common to study singular equivalences induced by suitable tensor product functors.

More specifically, we have the following definition from [Broue'1994]: Given a field \(k\) and two finite dimensional \(k\)–algebras \(A\) and \(B\), we say that a pair of bimodules \((_{B}M_{A},_{A}N_{B})\) defines a *stable equivalence of Morita type* between \(A\) and \(B\), if \(M\) (resp., \(N\)) is finitely generated and projective over \(B\) and \(A^{\mathrm{o}}\) (resp., over \(A\) and \(B^{\mathrm{o}}\)), and if the following hold:

\[N\otimes_{B}M\cong A\,\,\, \mbox{in}\,\,\, \underline{\mathrm{mod}}(A^e)\,\,\,\,\,\, \mbox{and}\,\,\,\,\,\, M\otimes_{A}N\cong B\,\,\, \mbox{in}\,\,\, \underline{\mathrm{mod}}(B^e).\]

Here \(A^{e}\), resp., \(B^{e}\), denotes the enveloping algebra of \(A\), resp., \(B\). In this situation there is an equivalence \(M\otimes_{A}-\colon \underline{\mathrm{mod}}(A)\rightarrow\underline{\mathrm{mod}}(B)\) with inverse \(N\otimes_{B}-\).

The above definition has has been generalized by Chen and Sun, and further by [Wang2015]. Wang's definition only differs than that of Broue' in that the conditions displayed above are now replaced by:

\[N\otimes_{B}M\cong \Omega_{A^{e}}^{l}(A)\,\,\, \mbox{in}\,\,\, \underline{\mathrm{mod}}(A^e)\,\,\,\,\,\, \mbox{and}\,\,\,\,\,\, M\otimes_{A}N\cong \Omega_{B^{e}}^{l}(B)\,\,\, \mbox{in}\,\,\, \underline{\mathrm{mod}}(B^e),\nonumber\]

where \(l\in\mathbb{N}\) and \(\Omega_{A^{e}}(-)\), resp., \(\Omega_{B^{e}}(-)\), denotes the syzygy endofunctor of the stable module category of \(A^{e}\), resp., \(B^{e}\). In this situation, there is a triangulated equivalence \(M\otimes_{A}-\colon \mathbf{D}_{\mathrm{sg}}(A)\rightarrow \mathbf{D}_{\mathrm{sg}}(B)\) with inverse \(\Sigma^{l}\circ(N\otimes_{B}-)\). Wang calls this *a singular equivalence of Morita type with level \(l\)* between \(A\) and \(B\).

In [Rickard1998] it is proved that for certain self-injective algebras, a stable equivalence induced from an exact functor is necessarily of Morita type (in the sense of Broue'). We are interested in an analogue of this for Gorenstein algebras. We record the main results:

Theorem Let \(A\) and \(B\) be two finite dimensional Gorenstein algebras such that \(A/\mathrm{rad}(A)\) and \(B/\mathrm{rad}(B)\) are separable. If there exists a complex \(X\) of finitely generated \(B\)-\(A^{\mathrm{o}}\)–bimodules which is perfect over \(B\) and \(A^{\mathrm{o}}\), and the functor \(X\otimes_{A}^\textbf{L}-\colon \mathbf{D}^{b}(\mathrm{mod}A)\rightarrow \mathbf{D}^{b}(\mathrm{mod}B)\) restricts to a singular equivalence, then it induces a singular equivalence of Morita type with level.

In terms of stable categories of (maximal) Cohen-Macaulay modules we obtain the following:

Corollary Let \(A\) and \(B\) be two finite dimensional Gorenstein algebras such that \(A/\mathrm{rad}(A)\) and \(B/\mathrm{rad}(B)\) are separable. If there exists a finitely generated \(B\)-\(A^{\mathrm{o}}\)–bimodule \(M\) which is projective on both sides and the functor \(M\otimes_{A}-\colon\mathrm{MCM}(A) \rightarrow\mathrm{MCM}(B)\) restricts to a triangulated equivalence \(\underline{\mathrm{MCM}}(A)\cong\underline{\mathrm{MCM}}(B)\), then the pair \((M,\Omega^{s}_{A\otimes_{k}B^{\mathrm{o}}}M^{\vee})\) defines a singular equivalence of Morita type with level \(s:=2\max\{\mathrm{Gdim}A,\mathrm{Gdim}B\}\) (where \(M^{\vee}:=\operatorname{Hom}_{B}(M,B)\)).

Note that the Corollary in the self-injective case recovers the aforementioned result of Rickard."

Tiago Cruz (University of Stuttgart)

A finite dimensional algebra is said to have dominant dimension at least \(n\in \mathbb{N}\) if there exists an exact sequence

\[0\rightarrow {}_AA\rightarrow I_1\rightarrow \cdots \rightarrow I_n\] with each \(I_i\) being a projective-injective \(A\)-module. Assume that \(A\) has dominant dimension at least one and let \(P\) be a faithful projective-injective module. The *Nakayama conjecture* asserts that if the dominant dimension is infinite then \(A\) is self-injective.

In such a case, \(\operatorname{End}_A(P)^{op}\) is also self-injective.

So, it is natural to ask:

(Q) When is \(\operatorname{End}_A(P)^{op}\) self-injective?

There are many positive examples to Q (e.g. Schur algebras, Morita algebras (see KernerYamagata2013)).

However, it is not always self-injective. For example, a negative answer for this can be obtained using the \(A_3\) quiver bound by the composition of the two arrows.

The main theorem of Cruz2020 is a characterisation for the algebras that give a positive answer to Q in terms of covers (in the sense of Rouquier2008). For its proof, a reformulation of Morita algebras using the Nakayama functor is also given.

More precisely,

Main Theorem

Let \(A\) be a finite-dimensional algebra. Assume that \(P\) is a faithful projective-injective left \(A\)-module. Then the following assertions are equivalent:
	a) \((A, P)\) is a cover of \(\operatorname{End}_A(P)^{op}\);
	b) \(A\) is a Morita algebra;
	c) The endomorphism algebra \(\operatorname{End}_A(P)^{op}\) is a self-injective algebra.

In particular, this theorem says that, for a projective-injective faithful \(A\)-module \(P\), the module \(\operatorname{Hom}_A(P, A)\) having a double centralizer property implies that \(\operatorname{End}_A(P)^{op}\) is self-injective. This reinforces that, in general, \(\operatorname{Hom}_A(P, A)\) is not necessairly injective even if \(P\) is projective-injective."

Gyoda, Yasuaki (Nagoya University)

This is joint work with Changjian Fu (Sichuan University). This article is based on [FuGyoda2019].

Let \(k\) be an algebraically closed field, and \(\mathcal{C}\) a Hom-finite \(2\)-Calabi-Yau triangulated category over \(k\) with suspension functor \(\Sigma\). In particular, for any \(X,Y\in \mathcal{C}\), we have the following bifunctorially isomorphism \[\text{Hom}_\mathcal{C} (X,Y)\cong D\text{Ext}^2_\mathcal{C}(Y, X),\] where \(D\) is the standard duality. An object \(M\in \mathcal{C}\) is *rigid* if \(\text{Ext}^1_\mathcal{C}(M,M)=0\). Furthermore, an object \(T\in\mathcal{C}\) is a *cluster-tilting object* if \(T\) is rigid and \(\text{Ext}^1_\mathcal{C}(T,Y)=0\) implies that \(Y\in \text{add}\ T\). For a basic cluster-tilting object \(T=U\oplus X\) with \(X\) indecomposable in \(\mathcal{C}\), there exists an indecomposable rigid object \(X'\) such that \(T'=U\oplus X'\) is a basic cluster-tilting object in \(\mathcal C\). Furthermore, \(X'\) is uniquely determined by the *exchange triangles* \[X\xrightarrow{f}B\xrightarrow{g}X'\to \Sigma X~\text{and}~X'\xrightarrow{f'}B'\xrightarrow{g'}X\to \Sigma X',\] where \(g, g'\) are minimal right \(\text{add}\ T\backslash X\)-approximations [IyamaYoshino2008]. The cluster-tilting object \(T'\) is called the *mutation* of \(T\) at the indecomposable direct summand \(X\) and we denoted by \(T'=\mu_X(T)\). In this case, \((X, X')\) is called an *exchange pair*. In parallel with mutation of cluster-tilting object, we define *Fomin-Zelevinsky's mutation* \(\mu_k\) of quiver. Let a quiver \(Q\) with no loops and no \(2\)-cycles. we denote by \(\{1,...,n\}\) the vertex set of \(Q\). We obtain \(Q'=\mu_k(Q)\) from \(Q\) as follows:

1. for each subquiver \(i\to k\to j\) in \(Q\), we add a new arrow \(i\to j\); 2. we reverse all arrows with source or target \(k\); 3. we remove the arrows in a maximal set of pairwise disjoint \(2\)-cycles.

For example, the following transformation is one of the Fomin-Zelevinsky's mutation.

!quiver|690x73, 50%

For a \(2\)-Calabi-Yau triangukated category \(\mathcal{C}\) with cluster-tilting objects, we consider the following conditions:

* for each basic cluster-tilting object \(T\), the Gabriel quiver \(Q_T\) of its endomorphism algebra \(\text{End}_\mathcal{C}(T)\) has no loops and no \(2\)-cycles; * if \(T=T\oplus X\) is a basic cluster-tilting object with \(X\) indecomposable, then \(Q_{\mu_X(T)}\) is the Fomin-Zelevinsky's mutation of \(Q_T\) at the vertex corresponding to \(X\).

If \(\mathcal C\) satisfies these conditions, then we say that \(\mathcal C\) admits a *cluster structure*. The main result is a characterization of exchange pair by the dimension of Ext-space:

Theorem. Let \(\mathcal{C}\) be a \(2\)-Calabi-Yau triangulated category with cluster-tilting objects, and \(X, X'\) two indecomposable rigid objects of \(\mathcal{C}\). If \(\dim_k\text{Ext}_{\mathcal{C}}^1(X, X')=1\), then \((X, X')\) is an exchange pair. Assume moreover that \(\mathcal{C}\) admits a cluster structure, then \(\dim_k\text{Ext}_{\mathcal{C}}^1(X, X')=1\) if and only if \((X, X')\) is an exchange pair.

Theorem can be regarded as an extension of the *exchangeablity property* of the compatibility degree associated with of generalized associahedra. That is, for a finite root system \(\Phi\) and any pair of almost positive roots \(\alpha\) and \(\beta\) in \(\Phi\), \((\alpha\parallel\beta)=(\beta\parallel\alpha)=1\) if and only if \(\alpha\) and \(\beta\) is exchangeable (for details of the compatibility degree of generalized associahedra, see [FominZelevinsky2003A] and [FominZelevinsky2003B])."

"A Counterexample to the \(\phi\)-dimension conjecture

Eric J. Hanson (Brandeis University) *Based on [HansonIgusa19+], joint work with Kiyoshi Igusa*

The \(\phi\)-dimension. Let \(\Lambda\) be a finite dimensional algebra over an arbitrary field \(K\) and let \(\mathsf{mod}\Lambda\) denote the category of finitely generated right \(\Lambda\)-modules. To study the famed finitistic dimension conjecture of [Bass60], the \(\phi\)-function is introduced in [IgusaTodorov05]. Given \(M \in \mathsf{mod}\Lambda\), we denote by \(\langle\mathsf{add} M\rangle\) the subgroup of the split Grothendieck group generated by the direct summands of \(M\). Given an integer \(t\), we define \(L^t(M)\) to be the rank of the (free abelian) group \(\langle \Omega^t M\rangle\), where \(\Omega\) is the syzygy functor. The quantity \(\phi(M)\) is defined to be the integer at which the sequence \((L^t(M))_t\) stabilizes.

The \(\phi\)-dimension is the supremum of the \(\phi\)-function over \(\operatorname{mod}\Lambda\). It is known to give an upper bound on the finitistic dimension and was conjectured to always be finite in [FernandesLanzilottaMendoza15]. We have shown that this conjecture is false. Another independent counterexample appeared in [BarriosMata19+] shortly later.

Our counterexample. Let \(K\) be a field and let \(A = KQ/R_Q^2\) and \(A_3^{CT} = KC_3/R_{C_3}^2\), where \(Q\) and \(C_3\) are the quivers shown below.

!Screen Shot 2020-10-28 at 6.22.17 PM|606x176

BlockquoteTheorem. The \(\phi\)-dimension of \(A \otimes_K A_3^{CT}\) is infinite.

Idea of the proof. Modules over \(A\otimes_K A_3^{CT}\) are equivalent to chain complexes over \(\mathsf{mod}A\) which are 3-periodic. Thus we find sequences \((X_k)_k\) and \((Y_k)_k\) of chain complexes over \(\mathsf{mod}A\) for which \(\Omega^{3k+1} X_k \cong \Omega^{3k+1} Y_k\), and \(3k+1\) is the smallest integer for which this holds. We then ""wrap"" \(X_k\) and \(Y_k\) around a 3-cycle so that they can be considered as modules over \(A \otimes_K A_3^{CT}\) and show that \(3k+1\) is still the smallest integer making their iterated syzygies isomorphic. This will imply that \(\phi(X_k \oplus Y_k) \geq 3k+1\).

The complexes \(X_k\) and \(Y_k\) are formed by ""truncating"" the projective resolutions of the simples (in \(\mathsf{mod} A\)) \(S_3\) and \(S_4\). For example:

!Screen Shot 2020-10-28 at 6.22.37 PM|690x175

and the complexes \(Y_k\) and \(\Omega Y_k\) are obtained by interchanging all indices 3 and 4.

If we index our complexes so that the terms furthest left are in degree \(-1\), then the ""rightmost discrepancy"" between \(X_1\) and \(Y_1\) is the map \(P_1 \rightarrow P_4\oplus S_3\) between degrees 4 and 3. On the other hand, the ""rightmost discrepancy"" between \(\Omega X_1\) and \(\Omega Y_1\) is the map \(S_1\oplus P_1\rightarrow P_3\oplus P_4\) between degrees 3 and 2. The discrepancy has moved to the left! This will continue until it ""falls off"" in degree -1.

The general complexes \(X_k\) and \(Y_k\) are defined so that their rightmost discrepancy is between degree \(3k\) and \(3k+1\)."

## 1. Partial morphisms in module categories

Our starting point is the notion of partial morphism in the category of right modules over a ring \(R\), which was introduced by Ziegler in his famous article *Model theory of modules* [Ziegler84]. He introduced it using model theoretic language:

—

### Definition

Given \(M\) and \(N\) modules, a *partial morphism* \(f\) from \(M\) to \(N\) is a \(R\)-linear map \(f:K \rightarrow N\), where \(K\) is a submodule of \(M\), such that, if \(\varphi\) is a pp-formula, then

\(M \models \varphi(a) \Rightarrow N \models \varphi(f(a))\)

The partial morphism \(f\) is called *partial isomorphism* if the converse of the preceding implication also holds.

—

Later, Monari-Martinez in [Monari84] characterized these notions in terms of systems of equations:

—

### Proposition

If \(M\) and \(N\) are modules, \(K\) is a submodule of \(M\) and \(f:K \rightarrow N\) is a morphism, then:

1) \(f\) is a partial morphism from \(M\) to \(N\) if and only if for every system of linear equations over \(K\),

\(\sum_{i=1}^nX_ir_{ij}=k_j \quad 1 \leq j \leq m,\)

which is solvable in \(M\), the system

\(\sum_{i=1}^nX_ir_{ij}=f(k_j) \quad 1 \leq j \leq m\)

is solvable in \(N\).

2) \(f\) is a partial isomorphism if and only if the converse of the preceding implication also holds.

—

The first result of our research gives a categorical characterization of these morphisms. This characterization involves pushouts and pure monomorphisms:

—

#### *Theorem*

If \(M\) and \(N\) are modules, \(K\) is a submodule of \(M\) and \(f:K \rightarrow N\) is a morphism, then:

1) \(f\) is a partial morphism from \(M\) to \(N\) if and only if the pushout of \(f\) along the inclusion \(i:K \rightarrow M\),

!Diagrama|154x153, 100%

satisfies that \(\overline i\) is a pure monomorphism.

2) \(f\) is a partial isomorphism if and only if \(\overline f\) is a pure monomorphism as well.

## 2. Partial morphisms in exact categories

Let \((\mathcal A;\mathcal E)\) be an exact category and \(\mathcal F\) an exact substructure of \(\mathcal E\). We can use the preceding characterization to define \(\mathcal F\)-partial morphisms (resp. \(\mathcal F\)-partial isomorphisms) in the category \(\mathcal A\), just assuming that \(K\) is an admissible subobject of \(M\) (that is, the inclusion \(i:K \rightarrow M\) is an inflation) and imposing that the resulting morphism \(\overline i\) (resp. \(\overline i\) and \(\overline f\)) is an \(\mathcal F\)-inflation (resp. are \(\mathcal F\)-inflations).

We can extend many properties of Ziegler partial morphisms to this new setting. One remarkable result is:

—

### Theorem

An object \(A\) of \(\mathcal A\) is \(\mathcal F\)-injective if and only if every partial morphism from \(M\) to \(E\) extends to a morphism \(f:M \rightarrow E\).

—

We can use partial morphisms to study the different notions of approximations that can be considered in an exact category, and to determine the relationship between them. In this line, we prove:

—

### Theorem

Let \(f:X \rightarrow E\) be an \(\mathcal F\)-inflation with \(E\) a \(\mathcal F\)-injective object. Then, under certain hypotheses, the following are equivalent:

1) \(f\) is an \(\mathcal F\)-essential morphism (a natural extension of the classical notion of essential monomorphism).

2) \(f\) is a Ziegler \(\mathcal F\)-small morphism.

3) \(f\) is a left minimal morphism (any \(g\) satisfying \(gf=g\) is an isomorphism).

—

Moreover, we can reduce the problem of finding these approximations to deciding if certain exact categories have enough injectives.

Finally, it is interesting to point out that \(\mathcal F\)-partial morphisms are natural extensions of \(\mathcal F\)-cophantom morphisms, which are the dual of the \(\mathcal F\)-phantom morphisms considered by Fu, Guil, Herzog and Torrecillas in [FuGuilHerzogTorrecillas2013].

This is joint research with P. A. Guil Asensio, Berke Kalebogaz and Ashish K. Srivastava."

Christian Lomp (University of Porto) Can Hatipoglu (American University of the Middle East,)

(based on arxiv:2010.14192 )

Let \(K\) be a field of characteristic zero. A result by S.Donkin (1982) shows that locally finite representations of the group ring \(A=K[[https://zbmath.org/?q=an%3A0485.17003|G]\) with \(G\) a polycyclic-by-finite group are closed under taking injective hulls. For a finite dimensional Lie algebra \(\mathfrak{g}\) over \(K\), locally finite dimensional representations of \(A=U(\mathfrak{g})\) are closed under taking injective hulls if and only if \(\mathfrak{g}\) is solvable, which holds by results of [Donkin (1982)]] and Feldvoss (2006). This motivated us to examine which Noetherian (Hopf) algebras \(A\) satisfy the condition that their category of locally finite dimensional representations \(\operatorname{Loc}(A)\) is closed under taking injective hulls (and hence forms a stable, hereditary torsion theory in \(A-\operatorname{Mod}\)).

Recall that the finite dual of a \(K\)-algebra \(A\) is the (possibly trivial) coalgebra

\(A^\circ = \{f \in \operatorname{Hom}_K(A,K) : \exists \mbox{ ideal } I \subseteq A, I\subset \operatorname{Ker}(f) \mbox{ and } \operatorname{dim}(A/I)<\infty\}\)

which is always a locally finite representation of \(A\).

What we found were the following results, which generalise the above mentioned cases:

* For a Noetherian \(K\)-algebra \(A\) , \(\operatorname{Loc}(A)\) is essentially closed if and only if the finite dual \(A^\circ\) is a (left) injective \(A\)-module. This basically follows from the observation that \(\operatorname{Loc}(A)\) is equal to the category of (right) \(A^\circ\)-comodules. Hence the injectivity of \(A^\circ\) is a necessary condition, which we show is also sufficient. * For Ore extensions \(A=R[x;\sigma]\) or \(A=R[x;\delta]\) with \(\sigma\) an automorphism and \(\delta\) a derivation of a commutative affine \(K\)-algebra \(R\), \(\operatorname{Loc}(A)\) is always essentially closed. This can be proven by passing from a maximal ideal \(M\) of finite codimension of \(A\) to \(M\cap R\), which in most cases (using the commutativity of \(R\)) has a Noetherian Rees ring and implies that \(M\) contains an ideal satisfying the Artin-Rees property. A similar result holds for a general Ore extension \(A=R[x;\sigma,\delta]\) provided \(R\) is a commutative Hopf algebra and \(\delta\) is a \(\sigma\)-derivation.

For an affine Noetherian Hopf algebra \(A\) over \(K\), \(\operatorname{Loc}(A)\) is essentially closed if and only if the injective hull \(E(K\)) of the trivial \(A\)-module \(K\) is locally finite.

We can show that locally finite representations of an affine Noetherian Hopf algebra \(A\) over \(K\) are closed under injective hulls in any of the following situations: * \(A=R\#_{\sigma} H\) is a crossed product of an affine commutative Hopf algebra \(H\) and a normal Noetherian Hopf subalgebra \(R\) whose augmentation ideal \(R^+\) satisfies the Artin-Rees property. (This basically mimics the solvable Lie algebra case.) * \(A\) is a domain of Gelfand-Kirillov dimension \(\leq 2\) and \(\operatorname{Ext}_A^1(K,K)\neq 0\) (by going through the classification result provided by Goodearl-Zhang (2010)) * \(A\) is a connected graded Hopf algebra of finite Gelfand-Kriillov dimension (by using the recent result Zhou-Shen-Lu (2020), which shows that in this case \(A\) is an iterated Hopf Ore extension in the sense of Brown et al. (2015)."

Sefi Ladkani (University of Haifa)

A *triangular algebra* is a quotient of a path algebra of a quiver without oriented cycles by an admissible ideal. To such algebra \(\Lambda\) one can associate its *Coxeter polynomial* \(\phi_\Lambda(x) = \det(x C_\Lambda + C_\Lambda^T)\), where \(C_\Lambda\) is the Cartan matrix of \(\Lambda\). It is a monic, self-reciprocal (palindromic) polynomial which carries significant homological information; \(\phi_\Lambda\) is a derived invariant of \(\Lambda\) and moreover the coefficient of \(x\) in \(\phi_\Lambda\) is strongly related to the Hochschild cohomology of \(\Lambda\) [Happel1997]00195-4).

It is an interesting problem to compute the Coxeter polynomials of various families of triangular algebras given in combinatorial terms. This has been done for the families of canonical, supercanonical and extended canonical algebras, see the survey [Lenzing-delaPena2008]. A method to compute the Coxeter polynomials of algebras obtained by gluing two quivers at a vertex [Boldt1995]00004-W) allows to recursively compute them for path algebras of trees.

Given a triangular algebra \(\Lambda\), a vertex \(s\) in its quiver and a finite poset \(S\), we can insert \(S\) at \(s\) and appropriately modify the ideal of relations to obtain a new triangular algebra \(\Lambda_{s \leftarrow S}\). This construction generalizes previous constructions in the literature, such as that of supercanonical algebras [LenzingdelaPena2004], or lexicographic sum of posets [Ladkani2008].

Simple examples show that the Coxeter polynomials of the algebras \(\Lambda\), \(\Lambda^-=(1-e) \Lambda (1-e)\) (where \(e\) is the primitive idempotent corresponding to \(s\)) and (the incidence algebra of) \(S\) are not sufficient to determine that of \(\Lambda_{s \leftarrow S}\). However, we can prove the following:

Theorem. Let \(S\) be a poset. There are polynomials \(\phi^0_S,\phi^1_S\) such that for any triangular algebra \(\Lambda\) and any vertex \(s\) in its quiver,

\(\)

\phi_{\Lambda_{s \leftarrow S}} =

\phi_\Lambda \cdot \phi^0_S + \phi_{\Lambda^-} \cdot \phi^1_S .

\(\)

The polynomials \(\phi^0_S\) and \(\phi^1_S\) can be expressed in terms of the Coxeter polynomials of \(S\) and the poset \(\widehat{S}\) obtained from \(S\) by adding a unique maximal element, and thus can be seen as refinement of the Coxeter polynomial of \(S\).

This result has diverse consequences. We list some of them below:

1. We recover the formulas for the Coxeter polynomials of coalescence of trees and those of supercanonical algebras, obtained previously using different methods.

1. We show that the Coxeter polynomial of any ordinal sum of posets is independent on the order of summands.

1. We can also show that \(\phi^0_S = 0\) for any poset \(S\) which is an orientation of an Euclidean diagram of type \(\widetilde{A}\). As a consequence we produce new families of triangular algebras whose Coxeter polynomials are products of cyclotomic polynomials.

1. We show that by taking \(S\) to range over the finite linearly ordered sets, we obtain sequences of algebras \(\Lambda_0, \Lambda_1, \Lambda_2, \dots\) which form *interlaced towers of algebras* in the sense of [delaPena2015]. This allows to construct examples refuting the claims made in Theorem 2 of [delaPena2015] on the spectral radius and Mahler measure of Coxeter polynomials for such sequences."

Xiaofa Chen (USTC) Xiao-Wu Chen (USTC)

Let \(k\) be a field and \(A\), \(B\) be finite dimensional \(k\)-algebras. It is a well-known conjecture posed by J. Rickard whether any triangle equivalence between bounded derived categories \(\mathbf D^b(A)\) and \(\mathbf D^b(B)\) is standard, that is, isomorphic to the derived tensor functor \(X\otimes^{\mathbb L}_{A} -\) by a two-sided tilting complex \(_{B}X_{A}\).

This question has been confirmed in several cases, such as hereditary algebras, Fano algebras, triangular algebras, the algebra of dual numbers and derived-discrete algebras of finite global dimension.

We confirm this conjecture for algebras derived equivalent to a smooth projective scheme, using dg category theory. For dg categories \(\mathcal A\) and \(\mathcal B\) we denote by \(\mathrm{rep}(\mathcal A,\mathcal B)\) the full subcategory of \(\mathcal D(\mathcal B^{op}\otimes \mathcal A)\) consisting of dg bimodules \(M\) such that \(M(-,A)\) is a quasi-representable right dg \(\mathcal B\)-module for each \(A\) in \(\mathcal A\). Roughly speaking, a triangle functor \(F\) is called liftable if it is of the form \(H^0(\tilde{F})\) for some \(\tilde{F}\) in Hqe.

Proposition There is a triangle equivalence $\mathrm{rep}(\mathbf D^b _{dg}(A-\mathrm{mod}),\mathbf D^b_{ dg}(B-\mathrm{mod})) \rightarrow \{M\in \mathbf D(B\otimes A^{op})|M_A \ \mathrm{is} \ \mathrm{perfect}\}$ sending a quasi-functor \(X\) to \(X(B,A)\). As a consequence, a triangle functor \(F: \mathbf D^b(A{-}\mathrm{mod}) \rightarrow \mathbf D^b(B{-}\mathrm{mod})\) is liftable if and only if it is standard.

For an abelian category \(\mathcal C\), a triangle functor \(F:\mathbf D^b(\mathcal C)\rightarrow \mathbf D^b(\mathcal C)\) is called pseudo-identity if \(F(X)=X\) for each complex \(X\) and \(F|_{\Sigma^n\mathcal C}:\Sigma^n\mathcal C\rightarrow \Sigma^n\mathcal C\) is the identity functor for each \(n\). Proposition Let \(F:\mathbf D^b(A{-}\mathrm{mod}) \rightarrow \mathbf D^b(\mathcal C)\) be a triangle equivalence. Then there is a factorization \(F\simeq F_2 \circ F_1\) of triangle functors, where \(F_1:\mathbf D^b(A{-}\mathrm{mod}) \rightarrow \mathbf D^b(A{-}\mathrm{mod})\) is a pseudo-identity and \(F_2:\mathbf D^b(A{-}\mathrm{mod}) \rightarrow \mathbf D^b(\mathcal C)\) is a liftable equivalence.

For a triangle functor \(F\) between bounded derived categories of coherent sheaves over smooth projective schemes, it is known by Toën and LuntsSchnürer that \(F\) is of Fourier-Mukai type if and only if it is liftable. So we get our desired result

Theorem Let \(A\) and \(B\) be finite dimensional algebras. Assume there is a triangle equivalence between \(\mathbf D^b(A{-}\mathrm{mod})\) and \(\mathbf D^b(\mathrm{coh-}\mathbb X)\) with \(\mathbb X\) a smooth projective scheme. Then any triangle equivalence \(F:\mathbf D^b(A{-}\mathrm{mod})\rightarrow \mathbf D^b(B{-}\mathrm{mod})\) is standard."

Zhen Zhang (BNU). Joint work with Aaron Chan (Nagoya U.), Jing Guo (USTC), Yuming Liu (BNU), and Yu Ye (USTC), based on [GLYZ1, 2020] and [CLZ, 2020].

Koenig and Liu [2011] introduced simple-minded system (s.m.s. for short) in the stable category of any artin algebra. Roughly speaking, an s.m.s. of an artin algebra \(A\) is a family of objects in the stable category \(A\!\(-\)\!\underline{\mathrm{mod}}\) which satisfies orthogonality and a generating condition. Chan, Koenig and Liu [2015] noticed that, for an indecomposable, basic representation-finite self-injective algebra \(A\) (\(\ncong\!k\)) over an algebraically closed field \(k\) (RFS algebra for short), the s.m.s.'s in \(A\!\(-\)\!\underline{\mathrm{mod}}\) correspond exactly to the combinatorial configurations in the stable AR-quiver of \(A,\) a notion introduced by Riedtmann in the 1980's.

In general, it is hard to check the two conditions in the definition of an s.m.s., however we give an easy characterization of s.m.s.'s over RFS algebras.

Theorem 1. [GLYZ1, 2020]) Let \(A\) be an RFS algebra and \(\mathcal{S}\) a family of objects in \(A\!\(-\)\!\underline{\mathrm{mod}}.\) Then \(\mathcal{S}\) is an s.m.s. if and only if \(\mathcal{S}\) satisfies the following three conditions. (1) \(\mathcal{S}\) is a Hom-orthogonal system in \(A\!\(-\)\!\underline{\mathrm{mod}},\) that is, for any $S,T\in\mathcal{S},\( \)
{\rm \underline{Hom}}_A(S,T)\cong \left\{\begin{array}{ll} 0 & (S\neq T),
k & (S=T).\end{array}\right.$ (2) The cardinality of \(\mathcal{S}\) is equal to the number of non-isomorphic simple \(A\!\)-modules. (3) \(\mathcal{S}\) is Nakayama-stable, that is, the Nakayama functor \(D(A)\otimes_{A}-\) on \(A\!\(-\)\!\underline{\mathrm{mod}}\) permutes the objects of \(\mathcal{S}.\)

As an application, we present an explicit construction of s.m.s.'s over self-injective Nakayama algebras ([GLYZ, 2020]). Note that our construction is independent with Riedtmann's (1980) and Chan's (2015) implicit ones.

We also study some behaviors of s.m.s.'s on a quasi-tube over a general self-injective algebra. The definition of a quasi-tube states as follows.

Definition 2. (1) A component of AR-quiver of a self-injective algebra is called a quasi-tube of rank \(n\) (homogeneous tube, if \(n=1\!\)), if its stable part (by removing projective vertices) is of the form \(\mathbb{Z}A_\infty/\langle\tau^n\rangle,\) where integer \(n\geqslant 1.\) (2) We call a non-projective module \(X\) of a quasi-tube \(\mathcal{C}\) is of quasi-length \(n\), if \(X\) is in the position \(n\) of an infinite sectional path \(A_\infty = (1\to 2 \to 3 \to \cdots)\) in the stable part of \(\mathcal{C}\).

Malicki and Skowronski [2011] showed that there are at most \(n-1\) simple modules on a quasi-tube of rank \(n\) over a self-injective algebra. Erdmann and Kerner's result [2006] indicates that, for a quasi-tube \(\mathcal{C}\) of rank \(n\), no object of \(\mathcal{C}\) with quasi-length strictly larger than \(n\) is in an s.m.s. We generalize and strengthen the above results as follows.

Theorem 3. [CLZ, 2020] Let \(A\) be a self-injective algebra, \(\mathcal{C}\) a quasi-tube of rank \(n\) of \(A,\) and let \(\mathcal{S}\) be an s.m.s. Then (1) \(\mid \mathcal{S}\cap\mathcal{C}\!\mid< n.\) In particular, none of the indecomposable module in an s.m.s. of \(A\) lie in any homogeneous tube. (2) No object of \(\mathcal{C}\) with quasi-length \(\geq n\) belongs to \(\mathcal{S}.\)"

Severin Barmeier (Albert-Ludwigs-Universität Freiburg / Hausdorff Research Institute for Mathematics Bonn)

<i>This snapshot is based on the preprint [BarmeierWang20] joint with Zhengfang Wang.</i>

Path algebras of quivers with relations are ubiquitous throughout algebra and geometry. Depending on their origin, their associative deformations can often be understood from a representation theoretic or geometric point of view, e.g. * deformations of \(A\) correspond to deformations of the Abelian category \(\operatorname{mod} (A)\) [LowenVandenBergh06] * deformations of \(A = \Bbbk [x_1, \dotsc, x_n]\) correspond to quantizations of Poisson structures on affine <i>n</i>-space \(\mathbb A^n\) * for \(\mathcal T\) a tilting bundle on an algebraic variety \(X\), deformations of \(A = \operatorname{End} (\mathcal T) \simeq \Bbbk Q / I\) correspond to deformations of the Abelian category of coherent sheaves on \(X\) [Keller03] [LowenVandenBergh05].

Deformations of associative algebras were first studied in [Gerstenhaber64] by ""deforming"" the associative multiplication on an associative algebra \(A\) as a \(\Bbbk\)-linear map \(A \otimes_{\Bbbk} A \to A\). More precisely, Gerstenhaber defined a bilinear operation, the <i>Gerstenhaber bracket</i>, which endows the (shifted) Hochschild cochain complex with the structure of a DG Lie algebra — which is precisely the structure used in the Maurer–Cartan formalism of deformation theory.

From a practical perspective, the space \(\operatorname{Hom} (A^{\otimes 2}, A)\) quickly becomes too large for concrete computations. However, when the associative algebra can be written as the path algebra of a quiver with relations, one may take advantage of this description and ""replace"" the bar resolution (giving rise to the Hochschild cochain complex in the classical study of deformations) by a smaller resolution: [ChouhySolotar15] constructed such a resolution for any finite quiver \(Q\) and any ideal of relations \(I\) and for any choice of reduction system satisfying the diamond condition for the ideal \(I\). (Such a reduction system always exists and may often be ""guessed"" or computed inductively.)

By constructing a homotopy between the Chouhy–Solotar resolution and the (reduced) bar resolution, we obtain the following general result.

Theorem. Let \(\Bbbk\) be a field of characteristic 0. Let \(Q\) be any finite quiver (loops and oriented cycles are allowed) and let \(I\) be any two-sided ideal of relations.
Let \(R\) be any reduction system satisfying the diamond condition for the ideal \(I\). Then there is an equivalence of deformation problems between:
* deformations of the associative algebra \(A = \Bbbk Q / I\)
* deformations of the reduction system \(R\).

The latter are controlled by an explicit L_∞ algebra and can be described purely combinatorially.

The combinatorial description can be viewed as a ""formal"" version of the resolution of overlaps in Bergman's diamond lemma [Bergman78]90010-5).

This point of view can also be used to * study the (homotopy/gauge) equivalences between deformations and determine the Maurer–Cartan space * give conditions for the existence of an algebraization (which may be evaluated for any value of the parameters) * prove convergence of certain formal deformations, even when they do not admit any obvious algebraization.

To illustrate how this can work in practice: the Brauer tree algebra !Screenshot from 2020-10-27 19-24-03|690x89, 75% is a 10-dimensional associative algebra whence \(\dim_{\Bbbk} \operatorname{Hom} (A^{\otimes 2}, A) = 1000\). From the point of view of reduction systems, the space of deformation candidates is only 4-dimensional and it suffices to check a handful of equations to find that up to equivalence this algebra admits a 1-dimensional family of nontrivial deformations. (Deformations of these Brauer tree algebras played a central role in [MazorchukStroppel11] in the representation theory of \(\mathfrak{sl}_n\).)

Using this point of view, a wide range examples can be studied in full detail. For applications and examples, also from geometry, see [BarmeierWang20, §8–11]."

He Ping

This snapshot is based on the preprint HeZhouZhu20 joint with Yu Zhou and Bin Zhu.

To a punctured marked surface \(\mathbf{S}\) with an admissible partial triangulation \(\mathbf{T}\), we define a skew-gentle algebra \(\Lambda^\mathbf{T}\). Moreover, we prove that any skew-gentle algebra can be obtained in this way.

We consider those curves on \(\mathbf{S}\) which winds around any unmarked boundary component no more than two times, and which cuts out a local triangle when it consecutively crosses two arcs in \(\mathbf{T}\) (following BaurSimoes19). Any such curve is said to be permissible if neither it cuts out a once-punctured monogon by its self-intersection nor its completion is a proper power of a closed curve. We always choose those permissible curves such that the one-step clockwise moving of any of its endpoints on the boundary of \(\mathbf{S}\) changes intersections between \(\gamma\) and \(\mathbf{T}\). A tagged permissible curve \((\gamma,\kappa)\) is a permissible curve \(\gamma\) with a 0/1-map \(\kappa\) defining on the punctured endpoints of \(\gamma\). The one-step clockwise moving of \(\gamma\), together with the opposite choice of \(\kappa\), is said to be the tagged rotation of \((\gamma,\kappa)\) and is denoted by \(\rho(\gamma,\kappa)\). The main result is the following.

Theorem There is a bijection \(\) \begin{array}{cccc}

      M:&\mathbb{P}^{\times}(\mathbf{S})&\to&\mathcal{S}\\
&(\gamma,\kappa)&\mapsto&M(\gamma,\kappa)
\end{array}

\(\) from the set \(\mathbb{P}^{\times}(\mathbf{S})\) of tagged permissible curves to the set \(\mathcal{S}\) of certain indecomposable \(\Lambda^\mathbf{T}\)-modules, which contain all indecomposable \(\tau\)-rigid modules. Moreover, we have the following.

1. If \(M(\gamma,\kappa)\) is not projective, then \[\tau M(\gamma,\kappa)=M(\rho(\gamma,\kappa)),\] where \(\tau\) is the Auslander-Reiten translation.

2. For \(M_1=M(\gamma_1,\kappa_1)\) and \(M_2=M(\gamma_2,\kappa_2)\), we have \[\dim\operatorname{Hom}(M_1,\tau M_2)+\dim\operatorname{Hom}(M_2,\tau M_1)=\operatorname{Int}((\gamma_1,\kappa_1),(\gamma_2,\kappa_2)),\] where \(\operatorname{Int}\) denotes the intersection number between tagged permissible curves.

We remark that

1. when \(\mathbf{S}\) is unpunctured (where \(\Lambda^\mathbf{T}\) is gentle), the intersection number \(\operatorname{Int}\) is the usual intersection number of curves, and

2. we also define the black intersection number to describe the dimension of \(\operatorname{Hom}(M_1,\tau M_2)\), although the definition is a little technical.

As an application of the Int-dim formula, we interpret the support \(\tau\)-tilting \(A\)-modules via the maximal sets of non-crossing tagged permissible curves, which possibly includes (tagged) curves in \(\mathbf{T}\)."

Amit Shah (Newcastle University)

*Joint work with Souheila Hassoun and Sven-Ake Wegner*

Based on [HassounShahWegner2020].

Since the 1960s there has been much research on additive, non-abelian categories. This has led to the development of a spectrum of classes of categories ranging from pre-abelian, to integral, to quasi-abelian, to abelian. With a focus on functional analysis and representation theory, we determine to which classes certain concrete categories from these two research areas belong. The following diagram summarises the outcome:

!pic|689x274, 100%

A *pre-abelian* category is an additive category in which every morphism has a kernel and a cokernel. Within the class of pre-abelian categories, one defines the following notions. *Semi-abelian* categories are the ones in which the canonical morphism between the coimage and image is always both monic and epic, but not necessarily an isomorphism as one would expect in an abelian category. *Quasi-abelian* categories are the ones in which kernels are stable under pushout and cokernels are stable under pullback. On the other hand, *integral* categories are the ones in which monomorphisms are stable under pushout and epimorphisms are stable under pullback.

Despite recent progress on integral categories, which appears to be predominantly in algebra, the question if there exist integral categories that are not quasi-abelian seems to date to be open. We answer this positively.

Theorem. The category \(\mathsf{BOR}\) of bornological locally convex spaces is integral, yet not quasi-abelian.

With an idea communicated to the authors by J. Wengenroth, we also show:

Theorem. There exist semi-abelian categories that are neither integral nor quasi-abelian. For example, the product category \(\mathsf{BAN}\times\mathsf{BOR}\), where \( \mathsf{BAN}\) is the category of Banach spaces, is such an example.

Non-abelian categories appear in abundance in functional analysis and have applications for instance in the theory of partial differential equations. Indeed, as can be seen from the diagram above, most of the categories we study are quasi-abelian but not abelian. However, this is still enough intrinsic structure to conduct homological algebra as Schneiders [Schneiders1999] did. He also observed that on each quasi-abelian category the class of all kernel-cokernel pairs forms an exact structure in the sense of Quillen [Quillen1973]. In contrast to the internal structure of a category, like pre-, semi- and quasi-abelian, an exact structure is extrinsic.

Recently, Brüstle, Hassoun and Tattar [BHT2020] have considered additive categories with a mix of intrinsic and extrinsic structures. They consider categories with the *admissible intersection property*: pre-abelian categories equipped with an exact structure in which admissible monomorphisms are stable under pullback along admissible monomorphisms. Jointly with Brüstle and Tattar we give a new characterisation for quasi-abelian categories:

Theorem. A pre-abelian category is quasi-abelian if and only if it has admissible intersections."

Hipolito Treffinger ( Rheinische Friedrich-Wilhelms-Universität Bonn)

This is a recollection of the results in [Treffinger2019] and [SchrollTreffinger2020]. Part of this work is joint with Sibylle Schroll.

Fix \(A\) to be a basic finite dimensional algebra over a field \(k\) such that \(A\), as a module over itself, is isomorphic to \(A \cong \bigoplus_{i=1}^n P(i)\). In this note we denote by \(\tau\) the Auslander-Reiten translation in \(\text{mod} A\), the category of finitely generated right \(A\)-modules.

In the last decade, one of the most prolific areas of research in representation theory has been the so-called \(\tau\)-tilting theory, introduced by Adachi, Iyama and Reiten introduced in [AdachiIyamaReiten2014] as a completion of tilting theory with respect to the notion of mutation. This theory studies the so-called \(\tau\)-tilting pairs and their behaviour in \(\text{mod} A\).

A \(\tau\)-tilting pair is a pair \((M,P)\) where \(M\) is an \(A\)-module and \(P\) a projective \(A\)-module such that \(\text{Hom}_A(M, \tau M)=0\), \(\text{Hom}_A(P,M)=0\) and \(M \oplus P\) has exactly \(n\) isomorphism classes of indecomposable direct summands.

It has been shown that \(\tau\)-tilting pairs are connected with several aspects of the homological properties of \(\text{mod}A\) for any algebra \(A\). For instance, it has been shown in [AdachiIyamaReiten2014] that there is a one to one correspondence between \(\tau\)-tilting pairs and functorially finite torsion pairs in \(\text{mod}A\).

Also, it is known that a \(\tau\)-tilting pair \((M,P)\) is completely determined by its \(g\)-vector, an integer vector encoding the minimal projective resolution of \(M\) and \(P\), see for instance [AuslanderReiten1984][DehyKeller2008][AdachiIyamaReiten2014] .

Moreover, the \(g\)-vectors of the indecomposable direct summands of \(M\) and \(P\) form a basis of \(\mathbb{Z}^n\). Using this fact and inspired by the tropical duality of cluster algebras, Fu defined in [Fu2017] the \(c\)-vectors of \((M,P)\) as the columns of the matrix \((G^{-1}_{(M,P)})^T\), where \(G_{(M,P)}\) is the square matrix having as columns the \(g\)-vectors of the indecomposable direct summands of \(M\) and \(P\). Before we state our first result, we recall that we say that an \(A\)-module \(B\) is a brick if \(\operatorname{End}_A(B_i)\) is a division ring. Our first result is the following.

Theorem. Let \(A\) be an algebra and let \((M,P)\) be a \(\tau\)-tilting pair with \(c\)-matrix \(C_{(M,P)}\). Then there is a diagonal matrix \(D_A\) which is independent of \((M,P)\) such that for every column \(\mathbf{c}_i\) of \(C_{(M,P)}\) there exists a non zero integer \(m_i\) and a brick \(B_i\) such that and
\(\)

D_A[B_{i}] = m_i \mathbf{c}_i,

\(\)
where \([B_i]\) is the class of \(B_i\) in the Grothendieck group \(K_0(A)\) of \(A\). Moreover, \(m_i\) is positive if and only if \(B_i\) belongs to the torsion class associated to \((M,P)\). Similarly, \(m_i\) is negative if and only if \(B_i\) belongs to the torsion-free class associated to \((M,P)\).

In [DemonetIyamaJasso19] Demonet, Iyama and Jasso introduced the notion of \(\tau\)-tilting finite algebras, which are algebras having finitely many \(\tau\)-tilting pairs. They showed that an algebra \(A\) is \(\tau\)-tilting finite if and only if there are finitely many torsion classes in \(\text{mod}A\) if and only if there are finitely bricks in \(\text{mod}A\). As a consequence of our previous result, we show the following theorem.

Theorem. Let \(A\) be an algebra. Then \(A\) is \(\tau\)-tilting finite if and only if the length of all bricks in \(\text{mod}A\) is bounded."

Adam-Christiaan van Roosmalen (Hasselt University)

This research snapshot is based on joint work with Ruben Henrard.

Exact categories are a more flexible framework for homological algebra and \(K\)-theory than abelian categories. However, not all constructions of interest preserve the self-dual nature of an exact category. For example, the localization of an exact category at a left or right multiplicative system [HenrardvanRoosmalen2019] or a pretorsion class in an exact category [HenrardvanRoosmalen2020] naturally prefer one side over the other, and one is led to consider so-called *one-sided exact categories* (in the sense of [Rump2001] or [BazzoniCrivei2013], see below). Furthermore, some categories of interest in functional analysis or representation theory have a canonical one-sided exact structure, such as the category of complete Hausdorff locally convex spaces over \(\mathbb{R}\) or \(\mathbb{C}\) (see [HassounShahWegner2020]) or the category of glider representations (see [CaenepeelVanOystaeyen2020] and [HenrardvanRoosmalen2020]).

Our aim is to show that the theory of one-sided exact structure is not really weaker than the theory of two-sided exact structures.

To establish some terminology, consider an additive category \(\mathcal{A}\). A *conflation structure* on \(\mathcal{A}\) is a class of chosen kernel-cokernel pairs, called *conflations*, closed under isomorphisms. We refer to the kernel map in a conflation as an *inflation* and to the cokernel part as a *deflation*. An additive category with a conflation structure is called a *conflation category*.

A conflation category is called *left exact* if the following conditions are satisfied: * E0: for any \(A \in \mathcal{A}\), the map \(A \to 0\) is a deflation, * E1: the composition of deflations is a deflation, * E2: pullbacks along deflations exist and deflations are stable under base change.

The notion of a *right exact* category is dual. A conflation category that is both left and right exact is called *exact*. Left exact categories are closely related to Grothendieck pre-topologies, and have been studied from this point of view in, for example, [Rosenberg2011] and [KaledinLowen2015].

For a left or right exact category \(\mathcal{D}\), the (bounded) derived category can be constructed in a similar way as in the exact case (see [BazzoniCrivei2013]): as the Verdier quotient of the homotopy category \(\operatorname{{\mathbf{K}^b}}(\mathcal{D})\) by the triangulated subcategory of acyclic complexes \(\operatorname{{\mathbf{Ac}^b}}(\mathcal{D})\).

In addition to the above axioms, for an exact category, one often considers the following axiom: * E3: for any morphism \(p\colon Y \to Z\) admitting a kernel, if there is a morphism \(f\colon X \to Y\) such that \(pf\) is a deflation, then \(p\) is a deflation.

This axiom is referred to as the *obscure axiom* in [ThomasonTrobaugh1990] and is known to be superfluous in the setting of exact categories (see [Keller1990]), but it is not superfluous in the setting of left exact categories. Its role in the one-sided exact setting will be illustrated below.

The Gabriel-Quillen embedding shows that every exact category occurs as an extension-closed subcategory of an abelian category. One can exploit this fact to transfer homological properties from the abelian setting to the exact setting. For one-sided exact categories, a similar statement holds (see Rosenberg2011): a left exact category \(\mathcal{D}\) can be 2-universally embedding into an exact category \(\mathcal{D}^{\mathrm ex}\); this is the exact hull of \(\mathcal{D}\). The following theorem ([HenrardvanRoosmalen2019] and [HenrardvanRoosmalen2020]) shows that \(\mathcal{D}\) and \(\mathcal{D}^{\mathrm ex}\) satisfy similar properties.

Theorem. Let \(\mathcal{D}\) be a left exact category.
1. The derived embedding \(j\colon \operatorname{{\mathbf{D}^b}}(\mathcal{D}) \to \operatorname{{\mathbf{D}^b}}(\mathcal{D}^{\mathrm ex})\) is a triangle equivalence.
2. A sequence \(A \stackrel{i}{\rightarrow} B \stackrel{p}{\rightarrow} C\) in \(\mathcal{D}\) is a conflation in \(\mathcal{D}^{\mathrm ex}\) if and only if there is a \(D \in \mathcal{D}\) such that !Sequence|179x40, 100% is a conflation in \(\mathcal{D}\).
3. The category \(\mathcal{D}\) satisfies the obscure axiom E3 if and only if the following holds: every sequence \(A \rightarrow B \rightarrow C\) in \(\mathcal{D}\) is a conflation in \(\mathcal{D}\) if and only if it is a conflation in \(\mathcal{D}^{\mathrm ex}\).

This theorem allows one to establish properties of \(\mathcal{D}\) using the exact hull \(\mathcal{D}^{\mathrm ex}\): for example, it follows from this theorem the \(3 \times 3\) lemma holds in a left exact category satisfying the obscure axiom E3.

As the triangle equivalence \(\operatorname{{\mathbf{D}^b}}(\mathcal{D}) \to \operatorname{{\mathbf{D}^b}}(\mathcal{D}^{\mathrm ex})\) is compatible with an enhancement by stable \(\infty\)-categories, the equivalence respects additive and localizing invariants in the sense of [BlumbergGepnerTabuada2013] (such as non-connective \(K\)-theory)."

Benjamin Blanchette (Université de Sherbrooke) *joint work with Thomas Brüstle*

Persistent homology studies the evolution of homology through filtrations of given topological spaces. Filtrations indexed by totally ordered sets are well understood and there are strong computational tools to work with them. For filtrations indexed by arbitrary posets, the theory is still at its inception. Algebraically, the maps induced on homology groups yield a representation of the given poset. The representation theory of arbitrary posets is typically wild, making the complete understanding of indecomposables impossible. Finding incomplete but rich invariants is therefore the more reasonable goal persistence homology researchers started working on. Our goal is to introduce many such invariants using results from relative homology, as well as describe established invariants using this algebraic approach.

The idea is to use various types of approximations as invariants. The usual approach is to approximate a representation by its projective cover. This is widely used in homological algebra, leveraging the knowledge we have on projective representations to study arbitrary representations. We apply the same principle using a larger family of indecomposables.

The theoretical framework is that of relative homology. Take \(X\) any finite set of indecomposable representations containing all projectives. A classical result of Auslander and Solberg (see AuslanderSolberg93, Theorem 1.15) establishes a bijection between additive subfunctors of \(Ext^1(-,-)\) with enough projectives and contravariantly finite generators of \(mod A\). \(X\) is such a finite generator, and the bijection yields an additive subfunctor \(F_X\) of \(Ext^1(-,-)\), which in turn corresponds to a weakly exact structure \(W_X\) in the sense of BaillargeonBrüstleGorskyHassoun20. We study minimal resolutions relative to \(X\), that is short exact sequences in \(W_X\) that combine to an exact sequence \[... \to X_2 \to X_1 \to M \to 0 \]

where each \(X_i\in X\) and every morphism is right minimal. The idea is essentially to cover each indecomposable factor using the indecomposables of \(X\) closest to it in the Auslander-Reiten quiver.

Consider the Grothendieck group relative to \(X\), that is the abelian group generated by isoclasses of representations quotiented by the relations induced by short exact sequences in \(W_X\). We are interested in two particular invariants arising from such a resolution.

The first is the \(g\)-vector, given by \([X_1]-[X_2]\).

The second one is the class of \(M\) in the Grothendieck group relative to \(X\), given by \(\sum_i (-1)^i [X_i]\).

Note that computations of those two present different challenges; the former requires a minimal but partial resolution, while the latter requires a complete but not necessarily minimal resolution. \[\] The poset given by the product of two totally ordered sets is of particular interest for applications; these are the representations associated with filtrations indexed by two real parameters, also referred to as 2D-persistence modules. A lot of recent work in persistence theory is using an algebraic approach toward studying representations of this poset. Notably, the recent paper AsashibaBuchetEscolarNakashimaYoshiwaki18 characterized thin indecomposables of compact support; they are thin indecomposables with identity maps over its support, which is a connected and convex subset of P. They are called interval representations and form a set of representatives for isoclasses of thin representations of this particular poset.

In a more recent paper ( see AsashibaEscolarNakashimaYoshiwaki19 ), the same authors introduced an invariant using this family of indecomposables. They call it an approximation by compressed multiplicities.

Along with developing computational procedures to make our invariant effective, we have two main conjectures:

Conjecture 1: The invariant given by taking the class of \(M\) in the Grothendieck group relative to interval representations is equivalent to the approximation by compressed multiplicities.

Conjecture 2: Every representation \(M\) admits a short interval resolution \(0\to I\to J\to M\to 0\).

We are also investigating the application of this constructions to arbitrary posets since intervals are well defined in the general case as well."

"<h1>Hopf actions of some quantum groups on path algebras</h1>

Amrei Oswald (University of Iowa)

Joint work with Ryan Kinser (University of Iowa) [KinserOswald2020]

Much like group actions formalize the notion of symmetry, Hopf actions of quantum groups formalize the notion of quantum symmetry. We analyze new examples of quantum symmetry by studying Hopf actions of the quantum groups \(U_q(\mathfrak{b})\), \(U_q(\mathfrak{sl}_2)\), generalized Taft algebras \(T(r,n)\), and the small quantum group \(u_q(\mathfrak{sl}_2)\) on path algebras.

Fix a field \(\Bbbk\) and assume \(\Bbbk\) contains any necessary roots of unity. Fix \(q \in \Bbbk^{\times}\) with \(q \neq \pm 1\) and two numbers \(m\) and \(m' \in \mathbb{Z}_{\geq 0}\). Let \(d=\text{gcd}(m,m')\) and \(\ell = \text{lcm}(m,m')\). We begin by parametrizing filtered actions of \(U_q(\mathfrak{b})\) and \(U_q(\mathfrak{sl}_2)\) on path algebras \(\Bbbk Q\) of a finite quiver \(Q\) using linear algebraic data.

Theorem. The following data determines a (filtered) Hopf action of \(U_q(\mathfrak{b})\) (resp., \(U_q(\mathfrak{sl}_2)\)) on a path algebra \(\Bbbk Q\), and all such actions are of this form:
(i) a Hopf action of \(U_q(\mathfrak{b})\) (resp., \(U_q(\mathfrak{sl}_2)\)) on \(\Bbbk Q_0\),
(ii) a representation of \(G\) on \(\Bbbk Q_1\) which is compatible with its \(\Bbbk Q_0\)-bimodule structure;
(iii) a linear endomorphism (resp. pair of linear endomorphisms) of \(\Bbbk Q_0 \oplus \Bbbk Q_1\)
satisfying certain conditions.

The linear endomorphism(s) mentioned in (iii) are derived from the actions of the standard skew-primitive generators of \(U_q(\mathfrak{b})\) and \(U_q(\mathfrak{sl}_2)\), but are in some sense independent of Hopf action on \(\Bbbk Q_0\) in (i). These parametrizations can be used to derive easily verifiable criteria for such actions to factor through the finite dimensional quotients \(T(r,n)\) and \(u_q(\mathfrak{sl}_2)\).

Following [EtingofKinserWalton2020], we view a path algebra with an action of a Hopf algebra \(H\) as particular case of a tensor algebra in the tensor category \(\mathsf{rep}(H)\). We wish to investigate the ""building blocks"" of actions of \(H\) on path algebras by describing minimal, faithful tensor algebras in \(\mathsf{rep}(H)\). To do so, we use the parametrization above to construct equivalences between categories of certain bimodules in \(\mathsf{rep}(H)\) and subcategories of finite-dimensional representations of associative \(\Bbbk\)-algebras, explicitly given in terms of quivers with relations below the theorem statement.

Theorem. Consider the tensor category \(\mathcal{C}=\mathsf{rep}(H)\) and a pair of indecomposable algebras \(S,\, S'\) in \(\mathcal{C}\) such that \(S=\Bbbk^m\) and \(S'=\Bbbk^{m'}\) as \(\Bbbk\)-algebras. Then the bimodule category \(\mathsf{bimod}_{\mathcal{C}}(S, S')\) is equivalent to the following category of representations in each case:
(a) for \(H=U_q(\mathfrak{b})\), representations of \(\Gamma(q^\ell,d)\) which are nilpotent on loops
(b) for \(H=T(r,n)\), representations of \(\Gamma_T\)
\(c\) for \(H=U_q(\mathfrak{sl}_2)\), representations of \(\Gamma'(q^{2\ell},q^2,d)\) which are nilpotent on loops
(d) for \(H=u_q(\mathfrak{sl}_2)\), representations of \(\Gamma'_T\)
where the algebras are defined below.

Let \(\mathcal{Q}\) be the quiver with vertex set \(\Bbbk^\times \times \mathbb{Z}/d\mathbb{Z}\). Coming out of each vertex \((\lambda,k)\) there is a loop and an arrow \((\lambda,k) \to (q^{-\ell}\lambda,k-1)\). Then, \(\Gamma(q^\ell,d)\) is the quotient of \(\Bbbk\mathcal{Q}\) with the ideal generated by all relations of the form \(ba = q^\ell ab\). The connected components of \(\mathcal{Q}\) have the form !Sinfinity|690x95, 75% when \(q\) is not a root of unity and !Sp|690x126, 75% when \(q\) is a root of unity where \(p = \text{lcm}(|q^\ell|,d)\). Therefore, it is not be feasible to give any kind of "list" classifying indecomposable \(S_m\(-\)S_{m'}\) bimodules in \(\mathsf{rep}(U_q(\mathfrak{b}))\), since the associated algebra \(\Gamma(q^\ell, d)\) will always have finite-dimensional quotients of wild representation type.

Say \(q\) is an \(r^{\text{th}}\) root of unity. Let \(\mathcal{T}\) be the quiver with vertices \((\zeta, i)\) and arrows \((\zeta,i+1)\to(\zeta,i)\) where \(\zeta\) is an \((n/\ell)^{\text{th}}\) root of 1 and \(i \in \mathbb{Z}/d\mathbb{Z}\). Define \(\Gamma_T\) to be the quotient of \(\Bbbk \mathcal{T}\) by the ideals generated by relations of the form \(a^r = 0\) if \(m\neq r \neq m'\) and \(a^t =\gamma_1\) for \(m = r\) or \(m'=r\) where \(t\) is the number so that \(r = td\) and \(\gamma_1\) is a constant. In the second case, \(\mathcal{T}\) has connected components of the following form. !taft|690x135, 75% It is easy to see that \(\Gamma_T\) is always finite dimensional over \(\Bbbk\). When the relations defining \(\Gamma_T\) only set paths to 0, it is a self-injective Nakayama algebra. The indecomposable objects in this situation can easily be described explicitly.

Now, we assume \(m, m' > 2\), and that \(q\) is a primitive \(n^{\rm th}\) root of unity with \(n>2\) and \(n\) odd. Let \(\mathcal{Q}'\) have the same vertex set as \(\mathcal{Q}\) with a loop at each vertex and arrows \((\lambda, k) \to (q^{-2\ell}\lambda,k-1)\) and \((\lambda, k) \to (q^{2\ell}\lambda,k-1)\). Then \(\Gamma'(q^{2\ell}\lambda,q^2,d)\) is the quotient of \(\Bbbk \mathcal{Q}'\) with the ideal generated by relations of the form \(ba = q^{2\ell}ab\), \(bc = q^{-2\ell} cb\), and \(q^2ac = ca\). An example of a connected component of \(\mathcal{Q}'\) where \(\mu\) is a third root of unity and \(d=3\) is shown below. !Uqsl2|690x273, 75% The quiver \(\mathcal{T}'\) is shown below. Then, \(\Gamma'_T\) is the quotient of \(\Bbbk \mathcal{T}'\) with the ideal generated by all relations of the form \(q^2ac = ca\), \(a^d = \gamma_2\), and \(c^d = \gamma_3\) where the \(\gamma_i\)'s are constants. A standard reordering argument shows that \(\Gamma'_T\) is always finite dimensional over \(\Bbbk\). !smallquantum|690x147, 75%"

Teresa Conde (University of Stuttgart) *This research snapshot is based on the preprint [Conde2020].*

Exact Borel subalgebras of quasihereditary algebras emulate the properties of 'classic' Borel subalgebras of complex semisimple Lie algebras.

Definition [Koenig1995] A subalgebra \(B\) of a quasihereditary algebra \((A,\Phi, \unlhd)\) is an *exact Borel subalgebra* if \(B\) is quasihereditary with respect to \((\Phi, \unlhd)\) with simple standard modules, and the induction functor \(A\otimes_B -\) is exact and maps the simples over \(B\) to the corresponding standard \(A\)-modules.

Not every quasihereditary algebra \((A,\Phi,\unlhd)\) has an exact Borel subalgebra. However, it was proved in [KoenigKülshammerOvsienko2014] that every quasihereditary algebra is *equivalent* to some quasihereditary algebra admitting an exact Borel subalgebra. Here, equivalence of quasihereditary algebras means that the corresponding categories of \(\Delta\)-filtered modules are equivalent. Equivalent quasihereditary algebras share the same quasihereditary structure and are, in particular, Morita equivalent.

Koenig, Külshammer and Ovsienko's existence theorem is quite remarkable and the exact Borel subalgebras in there can even be assumed to be homologically well behaved, i.e. *regular*. However, since their methods are abstract and rely on \(A_{\infty}\)- techniques, their work does not reveal how to explicitly construct a representative of a fixed quasihereditary algebra \((A,\Phi,\unlhd)\) containing an exact Borel subalgebra.

The goal in [Conde2020] is to answer the following questions, left unanswered in [KoenigKülshammerOvsienko2014]: A) *Given a quasihereditary algebra \((A,\Phi,\unlhd)\), how can we construct an equivalent algebra that contains a (regular) exact Borel subalgebra? Which algebras in the equivalence class \([(A,\Phi,\unlhd)]\) of \((A,\Phi,\unlhd)\) have a regular exact Borel subalgebra, and how many are there?* B) *How to decide whether a given quasihereditary algebra has a regular exact Borel subalgebra?* C) *In case \(B\) is a regular exact Borel subalgebra of some quasihereditary algebra equivalent to \((A,\Phi,\unlhd)\), is it possible to determine information about \(B\) (e.g. its Cartan matrix) without knowing \(B\) explicitly?* D) *When does it happen that every quasihereditary algebra equivalent to \([(A,\Phi,\unlhd)]\) contains a regular exact Borel subalgebra? When does a basic quasihereditary algebra admit a regular exact Borel subalgebra?*

Our strategy is as follows. To every quasihereditary algebra \((A,\Phi,\unlhd)\), we associate a special matrix \(V_{[(A,\Phi,\unlhd)]}=(v_{ij})_{i,j\in\Phi}\). The matrix \(V_{[(A,\Phi,\unlhd)]}\) can be computed through a recursive algorithm which takes as input the composition factors of the standard and costandard \(A\)-modules and also the dimension of the \(\operatorname{Hom}\)-spaces between standard \(A\)-modules. The distinguished matrix \(V_{[(A,\Phi,\unlhd)]}\) is therefore an invariant of the equivalence class \([(A,\Phi,\unlhd)]\) of \((A,\Phi,\unlhd)\). Furthermore, \(V_{[(A,\Phi,\unlhd)]}\) can be realised as a lower triangular matrix with nonnegative entries and with ones on the diagonal, hence \(V_{[(A,\Phi,\unlhd)]}\) is invertible. The sum of the entries in each row of \(V_{[(A,\Phi,\unlhd)]}=(v_{ij})_{i,j\in\Phi}\) shall be recorded in a sequence, denoted by \(l_{[(A,\Phi,\unlhd)]}=(l_{i})_{i\in\Phi}\).

Using the matrix \(V_{[(A,\Phi,\unlhd)]}=(v_{ij})_{i,j\in\Phi}\) and the sequence \(l_{[(A,\Phi,\unlhd)]}=(l_i)_{i\in\Phi}\), we answer the questions raised in A), B), C) and D) in Theorems A, B, C and D, respectively. The notation \(L_i=L_i^A\) will be used for the simples over \(A\) and \(\mathcal{F}(\Delta)\) (resp. \(\mathcal{F}(\nabla)\)) shall denote the category of \(\Delta\)-filtered modules (resp. \(\nabla\)-filtered modules).

Theorem A Let \((A,\Phi,\unlhd)\) be quasihereditary with projective indecomposable modules \(P_i\), \(i\in \Phi\).

1) For every sequence of positive integers \((k_i)_{i\in \Phi}\), the quasihereditary algebra \((\operatorname{End}(\bigoplus_{i\in\Phi}P_i^{m_i})^{op},\Phi,\unlhd)\), with \(m_i=\sum_{j\in\Phi} v_{ij}k_i\), is (up to isomorphism) the only algebra equivalent to \((A,\Phi,\unlhd)\) that contains a regular exact Borel subalgebra satisfying \(\dim L_i^B=k_i\) for every \(i\in\Phi\). 2) \((\operatorname{End}(\bigoplus_{i\in\Phi}P_i^{l_i})^{op},\Phi,\unlhd)\) is (up to isomorphism) the only quasihereditary algebra equivalent to \((A,\Phi,\unlhd)\) that has a basic regular exact Borel subalgebra.

Theorem B Let \((A,\Phi,\unlhd)\) be a quasihereditary algebra.

1) \((A,\Phi,\unlhd)\) has a regular exact Borel subalgebra if and only if the unique solution of the system \(V_{[(A,\Phi,\unlhd)]} x=(\dim L_i^A)_{i\in\Phi}\) is a vector whose entries are positive integers. 2) If \((A,\Phi,\unlhd)\) has a regular exact Borel subalgebra \(B\), then \((\dim L_i^B)_{i\in\Phi}\) is the only solution of the system \(V_{[(A,\Phi,\unlhd)]} x=(\dim L_i^A)_{i\in\Phi}\). 3) \((A,\Phi,\unlhd)\) has a basic regular exact Borel subalgebra if and only if \(\dim L_i^A=l_i\) for every \(i\in\Phi.\)

Theorem C Let \((A,\Phi,\unlhd)\) be a quasihereditary algebra. Assume that \(B\) is a regular exact Borel subalgebra of some quasihereditary algebra equivalent to \((A,\Phi,\unlhd)\). The Cartan matrix of \(B\) is given by \((D_{[(A,\Phi, \unlhd)]}^{\nabla} V_{[(A,\Phi, \unlhd)]})^T\), where \(D_{[(A,\Phi, \unlhd)]}^\nabla\) denotes the \(\nabla\)-decomposition matrix \(([\nabla_i^A:L_j^A])_{i,j\in\Phi}\) of \((A,\Phi,\unlhd)\). In particular, the Cartan matrix of \(B\) is completely determined by the composition factors of the standard and costandard \(A\)-modules and by the dimension of the \(\operatorname{Hom}\)-spaces between standard modules, so it only depends on \([(A,\Phi,\unlhd)]\).

Theorem D Let \((A,\Phi,\unlhd)\) be a quasihereditary algebra. The following conditions are equivalent:

1) every quasihereditary algebra equivalent to \((A,\Phi,\unlhd)\) has a regular exact Borel subalgebra; 2) the sequence \(l_{[(A,\Phi,\lhd)]}\) is constant and equal to one; 3) \(V_{[(A,\Phi,\lhd)]}\) is the identity matrix; 4) for every \(i \in \Phi\), \(\Delta_i\) is a right \(\mathcal{F}(\Delta)\)-approximation of the simple \(A\)-module \(L_i\) (meaning that every morphism \(X\rightarrow L_i\) with \(X\in \mathcal{F}(\Delta)\) factors through the epic \(\pi_i: \Delta_i \twoheadrightarrow L_i\)); 5) \(\operatorname{Rad}{\Delta_i}\) belongs to \(\mathcal{F}(\nabla)\) for every \(i \in \Phi\); 6) the map \(\operatorname{Ext}_{A}^{1}(X,\pi_i):\operatorname{Ext}_{A}^{1}(X,\Delta_i)\rightarrow\operatorname{Ext}_{A}^{1}(X,L_i)\), where \(\pi_i\) denotes the epic from \(\Delta_i\) to \(L_i\), is an isomorphism for every \(X\) in \(\mathcal{F}(\Delta)\) and every \(i\in\Phi\). Assuming that \(A\) is basic, then the algebra \((A,\Phi,\unlhd)\) contains a regular exact Borel subalgebra if and only if it contains a basic regular exact Borel subalgebra if and only if one of the equivalent conditions 1) to 6) holds."

Nan Gao (Shanghai University) Julian Külshammer (Uppsala University) Sondre Kvamme (Uppsala University) Chrysostomos Psaroudakis (Aristotle University of Thessaloniki)

*Partially based on arXiv:1907.04657.*

Let \(\Bbbk\) be a field and let \(Q\) be a finite (or locally bounded) quiver. Recently, there has been renewed interest in a generalisation of the path algebra \(\Bbbk Q\) called (generalised) species [Li12, GeissLeclercSchröer17]. Originally used to describe the representation theory of finite dimensional algebras over non-algebraically closed fields, the concept has recently been applied in the theory of cluster algebras. A generalised species \(\Lambda\) associated to a quiver \(Q\) is an association of a finite dimensional algebra \(\Lambda_{\mathtt{i}}\) to each vertex \(\mathtt{i}\in Q_0\) and a \(\Lambda_{t(\alpha)}\(-\)\Lambda_{s(\alpha)}\)-bimodule \(\Lambda_\alpha\), finitely generated projective from either side, to each arrow \(\alpha\in Q_1\) such that \(\) \operatorname{Hom}_{\Lambda_{s(\alpha)}^{\operatorname{op}}}(\Lambda_\alpha,\Lambda_{s(\alpha)}) \cong \operatorname{Hom}_{\Lambda_{t(\alpha)}}(\Lambda_\alpha,\Lambda_{t(\alpha)}) \(\) as \(\Lambda_{s(\alpha)}\(-\)\Lambda_{t(\alpha)}\)-bimodules . To a generalised species one can associated the tensor algebra \(T(\Lambda)=T_{\prod \Lambda_{\mathtt{i}}}(\bigoplus_\alpha \Lambda_\alpha)\). It is known to be \((n+1)\)-Iwanaga–Gorenstein if all the \(\Lambda_{\mathtt{i}}\) are \(n\)-Iwanaga–Gorenstein, see e.g. [Wang16]. Furthermore, a \(T(\Lambda)\)-module is Gorenstein projective if and only if the map \(\) M_{\mathtt{i},\operatorname{in}}\colon \bigoplus_{\alpha\colon s(\alpha)\to \mathtt{i}} \Lambda_\alpha\otimes_{\Lambda_{s(\alpha)}} M_{s(\alpha)}\stackrel{(M_\alpha)_\alpha}{\longrightarrow} M_{\mathtt{i}} \(\) is a monomorphism for all \(\mathtt{i}\in Q_0\) and the modules \(\operatorname{Coker}(M_{\mathtt{i}, \operatorname{in}})\) are all Gorenstein projective. The monomorphism category, where only the monomorphism condition is satisfied, is therefore an interesting `approximation' of the Gorenstein projective modules over an algebra. (Dually one can define the epimorphism category.) In the case that \(Q=A_2\) and \(\Lambda_\mathtt{1}=\Lambda_\mathtt{2}=A\) and \(\Lambda_\alpha=A\) it has been intensively studied by Ringel and Schmidmeier [RingelSchmidmeier06, RingelSchmidmeier08]. In joint work in progress we generalise their work to the case of an arbitrary generalised species. Analogously to the case of the path algebra, one can define a (relative) inverse Nakayama functor \(\nu^-\) whose image precisely coincides with the monomorphism category.

Theorem * Let \(0\to L\to M\to N\to 0\) be an almost split sequence in \(T(\Lambda)\) with \(N\) being in the epimorphism category, but not projective in there. Then \(0\to \nu^-(L)\to \nu^-(M)\to \nu^-(N)\to 0\) is a direct sum of an almost split sequence and a split sequence in the monomorphism category. * The Auslander–Reiten translation \(\tau\) in the monomorphism category can be computed as \(\tau=\operatorname{Mimo}\circ \left(\prod_{\mathtt{i}} \tau_{\Lambda_{\mathtt{i}}}\right)\circ \nu\) where \(\nu\) denotes the relative Nakayama functor, \(\prod_{\mathtt{i}} \tau_{\Lambda_{\mathtt{i}}}\) denotes the pointwise application of the Auslander–Reiten translation on \(\operatorname{mod} \Lambda_{\mathtt{i}}\) and \(\operatorname{Mimo}\) denotes the minimal right approximation to the monomorphism category (which admits an explicit description).

Our results are deduced from the general theory of relative Nakayama functors developed in [Kvamme20] in the following categorical context: Let \(\mathcal{C}\) be an abelian category (here \(\operatorname{mod} \prod \Lambda_\mathtt{i}\)). Let \(X\) and \(Y\) be endofunctors on \(\mathcal{C}\) (here e.g. \(X=\bigoplus_{\alpha} \Lambda_\alpha\otimes_{\prod \Lambda_{\mathtt{i}}}-\)) with the following properties:

* \((X,Y)\) is a pair of Frobenius functors, i.e. \(X\) is left as well as right adjoint to \(Y\). * The coproducts and products of powers of \(X\) exist and the natural map between them is an isomorphism. * There is no epimorphism \(X(M)\to M\) and no monomorphism \(M\to Y(M)\) for \(M\neq 0\).

In this setup, the monorphism category is the category of all monomorphism \(X(M)\to M\), which can be viewed to live in the Eilenberg–Moore category of the free monad on \(X\). In fact, quite a lot of the basic theory of representations of quivers can be generalised to this categorical setup, including the preprojective algebra."

Rose Wagstaffe (University of Manchester)

This Research Snapshot is based on the preprint [Wagstaffe2020].

Let \(\mathcal{C}\) be a finitely accessible category with products, for instance the category of modules over some ring. A full subcategory \(\mathcal{D} \subseteq \mathcal{C}\) is said to be definable if there exists a collection of finitely presented functors, \(\Phi \subseteq (\mathcal{C}^{\mathrm{fp}},\mathbf{Ab})^{\mathrm{fp}}\), such that \(\mathcal{D}=\{ X \in \mathcal{C}: F(X)=0,~\forall F \in \Phi \}.\) In a number of examples \(\mathcal{C}\) comes equipped with a monoidal structure \((\otimes, 1)\). For example, if \(R\) is a commutative ring, the tensor product of \(R\)-modules, \(\otimes_R\), provides \(R\hbox{-}\mathrm{Mod}\) with a closed additive symmetric monoidal structure. Thus, it is natural to ask how the definable subcategories might interact with the monoidal structure.

The definable subcategories of \(\mathcal{C}\) correspond via annihilation to the Serre subcategories of \(\mathcal{C}^{\mathrm{fp}}\hbox{-}\mathrm{mod}\) (see [Herzog1997] or [Krause1997]). Note that if \(\mathcal{C}^{\mathrm{fp}}\) is monoidal, we can induce a monoidal structure on \(\mathcal{C}^{\mathrm{fp}}\hbox{-}\mathrm{Mod}\) via Day convolution product [Day70]. Furthermore, the finitely presented \(\mathcal{C}^{\mathrm{fp}}\)-modules form a monoidal subcategory and if a Serre subcategory \(\mathsf{S}\) of \(\mathcal{C}^{\mathrm{fp}}\hbox{-}\mathrm{mod}\) is a tensor-ideal, then the abelian category \(\mathcal{C}^{\mathrm{fp}}\hbox{-}\mathrm{mod}/ \mathsf{S}\) has a unique monoidal structure such that the localization functor is monoidal. We introduce the following definition to characterise the definable subcategories which correspond to Serre tensor-ideals.

Definition Suppose \(\mathcal{D}\) is a definable subcategory of a finitely accessible category with products \(\mathcal{C}\). Suppose further, that \(\mathcal{C}\) has a closed additive symmetric monoidal structure and denote the internal hom-functor by \(\mathrm{hom}:\mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathcal{C}\). We say that \(\mathcal{D}\) is fp-hom-closed if for every \(C \in \mathcal{C}^{\mathrm{fp}}\) and \(X \in \mathcal{D}\), \(\mathrm{hom}(C, X) \in \mathcal{D}\).

Theorem Fix a finitely accessible category \(\mathcal{C}\) with product and a closed additive symmetric monoidal structure. Suppose further that the full subcategory of finitely presented objects \(\mathcal{C}^{\mathrm{fp}}\) forms a monoidal subcategory. A definable subcategory \(\mathcal{D} \subseteq \mathcal{C}\) is fp-hom-closed if and only if the corresponding Serre subcategory \(\mathsf{S} \subseteq \mathcal{C}^{\mathrm{fp}}\hbox{-}\mathrm{mod}\) is a tensor-ideal.

In [PrestRajani2010] a 2-category anti-equivalence is established between the 2-category \(\mathbb{ABEX}\) of (skeletally) small abelian categories and exact functors and the 2-category \(\mathbb{DEF}\) of definable categories and additive functors which commute with direct products and direct limits.

Definition Define \(\mathbb{ABEX}^{\otimes}\) to be the 2-category with objects given by (skeletally) small abelian categories with an additive symmetric monoidal structure which is exact in each variable, 1-morphisms given by exact monoidal functors and 2-morphisms given by natural transformations.

In [Wagstaffe2020] we define a 2-category \(\mathbb{DEF}^{\otimes}\) with objects given by definable subcategories \(\mathcal{D} \subseteq \mathcal{C}\) with some additional monoidal structure, such that there exists a 2-category anti-equivalence between \(\mathbb{ABEX}^{\otimes}\) and \(\mathbb{DEF}^{\otimes}\)."

András Cristian Lőrincz (Humboldt-Universität zu Berlin)

The operation of node splitting for finite-dimensional algebras was introduced in [Martı́nez-Villa80]. Let \(Q\) be a quiver with its path algebra \(\mathbb{k} Q\). A node of an algebra \(A=\mathbb{k} Q/I\) is a vertex \(x\) of \(Q\) such that all the paths of length \(2\) passing strictly through \(x\) belong to the ideal of relations \(I\). A node \(x\) of \(A\) can be split by the following local operation around \(x\) obtaining an algebra \(A^x=\mathbb{k} Q^x/ I^x\):

!node1|633x146, 75%

The representation theory of \(A\) and \(A^x\) are very closely related. Together with Ryan Kinser, in [KinserLőrincz18] we analyze the geometry behind node splitting. The set of representations with a fixed dimension vector \(\alpha\) form an affine variety \(\operatorname{Rep}(A,\alpha)\) that lives in a product of spaces of matrices, and has an action of a product of general linear groups corresponding to changing bases at vertices. We investigate natural varieties in this space (e.g. irreducible components, orbit closures).

We build a map between varieties of representations from \(A^x\) to \(A\) that preserves the property of normality and rational singularities, and relates their defining equations. The strength of our method lies in working in the relative setting, i.e. splitting locally a node without assuming any restrictions on the rest of the algebra.

While the scope of the results is much broader, they are easiest to state in the case when \(A\) is a radical square zero algebra, i.e. when every vertex is a node. To illustrate, if \(Q\) is the \(\mathbb{A}_{n+1}\) quiver !node|534x51, 100% then the relations are \(a_{i+1} \circ a_i=0\), and a representation of \(A\) is a complex. In fact, when \(Q\) is an equioriented type \(\mathbb{A}\) quiver, a loop, or a \(2\)-cycle, then these varieties of representations include the Buchsbaum–Eisenbud variety of complexes, variety of projectors, and the variety of circular complexes, respectively, which have been investigated extensively. Splitting nodes repeatedly, we sharpen and generalize the results to arbitrary quivers.

Theorem. Let \(A\) be a radical square zero algebra. Then for any dimension vector \(\alpha\), each irreducible component \(C \subseteq \operatorname{Rep}(A,\alpha)\) has rational singularities when \(\operatorname{char} \mathbb{k} =0\) (resp. is strongly \(F\)-regular when \(\operatorname{char} \mathbb{k} >0\)), and is thus normal and Cohen–Macaulay.

We further give explicit defining equations for such \(C\) in \(\operatorname{Rep}(\mathbb{k} Q,\alpha)\). By the work [ChindrisKinser18], the normality of the irreducible components of \(\operatorname{Rep}(A,\alpha)\) has important consequences for the decomposition of its moduli spaces of semistable representations.

Our main technique is that of collapsing of bundles, following [Kempf76]. Let \(G\) be a connected reductive group over an algebraically closed field. Consider a parabolic subgroup \(P\subset G\), let \(W\) be a \(G\)-module and \(V\subset W\) a \(P\)-stable submodule. The saturation \(G\cdot V\) is the image of the homogeneous vector bundle \(G \times_P V\) under the ""collapsing map"" \(G \times_P V \to W\) induced by the action of \(G\) on \(W\).

Kempf showed that in characteristic zero \(G\cdot V\) has rational singularities whenever the unipotent radical of \(P\) acts trivially on \(V\). In [Lőrincz20], I generalize Kempf's result, providing a characteristic-free strengthening, showing that \(G\cdot V\) is strongly \(F\)-regular under the presence of good filtrations, as introduced by Donkin. The techniques on singularities rely on tight closure theory, that was initiated by Hochster and Huneke. Furthermore, I provide a characteristic-free relative result on defining equations of saturations. I extend the technique to collapsing of bundles over Schubert varieties, generalizing, in particular, results on multicones over Schubert varieties and matrix Schubert varieties."

Job D. Rock (Hausdorff Research Institute for Mathematics) *Based on [IgusaR.Todorov2020, R.2020]. The first is joint work with Kiyosi Igusa and Gordana Todorov.*

### Motivation Cluster categories were introduced in [BuanMarshReinekeReitenTodorov2006, CalderoChapatonSchiffler2006]. There are generalizations to the \(\infty\)-gon [HolmJørgensen2012] and completed \(\infty\)-gon [BaurGratz2018]. There is a further generalization to discrete laminations of the hyperbolic plane [IgusaTodorov2015b] and a further algebraic generalization in [IgusaR.Todorov2020]. Geometrically, all of these cluster structures should be related. However, functors cannot preserve all the triangulated and cluster structures simultaneously.

### Cluster Theories and Abstract Cluster Structures Cluster theories and embeddings of cluster theories were introduced in [IgusaR.Todorov2020]. The ability to mutate *every* object in a cluster is not required. A unique choice of mutation, if it exists, is required. Abstract cluster structures and the corresponding embeddings are introduced in [R.2020].

""Definition"" Let \(\mathcal C\) be a Krull–Schmidt category and \(\mathbf P\) a pairwise compatibility condition on its (isoclasses of) indecomposables. *If it exists*, the cluster theory \(\mathscr T_{\mathbf P}(\mathcal C)\) is a groupoid in the category of sets with embedding \(I_{\mathbf P,\mathcal C}:\mathscr T_{\mathbf P}(\mathcal C)\to \mathcal Set\). Objects of \(\mathscr T_{\mathbf P}(\mathcal C)\) are maximally \(\mathbf P\)-compatible sets of indecomposables in \(\mathcal C\); morphisms are compositions of \(\mathbf P\)-mutations.
An embedding of cluster theories \((F,\eta):\mathscr T_{\mathbf P}(\mathcal C) \to\mathscr T_{\mathbf Q}(\mathcal D)\) is an embedding \(F:\mathscr T_{\mathbf P}(\mathcal C)\to \mathscr T_{\mathbf Q}(\mathcal D)\) and a natural transformation \(\eta:I_{\mathbf P,\mathcal C}\to I_{\mathbf Q, \mathcal D}\circ F\) whose component morphisms are injections.

""Definition"" *If it exists*, the abstract cluster structure \(\mathscr S_{\mathbf P}(\mathcal C)\) is a subgroupoid of \(\mathscr T_{\mathbf P}(\mathcal C)\). For each \(\mathbf P\)-cluster \(T\), every \(x\in T\) must be \(\mathbf P\)-mutable and \(\mathcal C / \operatorname{add}T\) must be abelian. Furthermore, \(\mathscr S_{\mathbf P}(\mathcal C)\) must be maximal with respect to these properties. Embeddings of abstract cluster structures are defined similarly to those for cluster theories.

Every Dynkin type cluster category yields a cluster theory. However, the pairwise compatibility conditions in [HolmJørgensen2012, IgusaTodorov2015b], for example, yield cluster theories, not cluster structures. The cluster structures examined by those authors are indeed abstract cluster structures.

### Main Results Let \(\mathbf N_m\) and \(\mathbf N_n\) be the pairwise compatibility conditions for \(A_m\) and \(A_n\), respectively, from [BuanMarshReinekeReitenTodorov2006, CalderoChapatonSchiffler2006]. Let \(\mathbf N_\infty\), \(\mathbf N_{\overline{\infty}}\), and \(\mathbf N_{\mathbb R}\) be the pairwise compatibility conditions in [HolmJørgensen2012], [BaurGratz2018], and [IgusaTodorov2015b], respectively. Finally, let \(\mathbf E\) be the pairwise compatibility condition introduced in [IgusaR.Todorov2020]. The cluster categories in the notation are suppressed in the theorems.

Theorem [R.2020] There is a chain of embeddings of cluster theories, for \(0<m<n\),
!Screen Shot 2020-10-26 at 00.07.36|690x48, 75%

Theorem [R.2020] There is a commutative diagram of embeddings of cluster theories (left) that restricts to a commutative diagram of embeddings of abstract cluster structures (right):
!Screen Shot 2020-10-26 at 00.03.52|690x143, 75%

### Outlook There are other type \(A\) cluster categories [IgusaTodorov2015a, PaquetteYıldırım2020]. Does including these cluster theories still yield a chain or something like a lattice instead? Do they have abstract cluster structures and if so how do they fit into the second theorem?"

Mads Hustad Sandøy (NTNU)

Joint with Johanne Haugland

While algebras of global dimension \(0\) and \(1\) are exceptionally well understood, it seems quite difficult to develop a general theory for algebras of higher global dimension. This is a background for studying the class of \(n\)-hereditary algebras Herschend-Iyama-Oppermann'2014 which plays an important role in higher Auslander-Reiten theory. An \(n\)-hereditary algebra has global dimension less than or equal to \(n\) and is either \(n\)-representation finite or \(n\)-representation infinite. The case \(n=1\) recovers the classical definitions of representation finite and infinite hereditary algebras.

Like in the classical theory, \(n\)-hereditary algebras have a notion of (higher) preprojective algebras. Using known equivalences of categories and the notation \(\Delta A\) for the trivial extension of \(A\), one obtains \[ \operatorname{\underline{gr}} (\Delta A) \simeq \operatorname{D}^b(\operatorname{mod} A) \simeq \operatorname{D}^b(\operatorname{qgr} \Pi_{n+1}A).\]

The equivalence above brings to mind the acclaimed Bernstein-Gelfand-Gelfand-correspondence, which can be formulated as \(\operatorname{\underline{gr}} \Lambda \simeq \operatorname{D}^b(\operatorname{qgr} \Lambda^!)\) for a finite dimensional Frobenius Koszul algebra \(\Lambda\) and its graded coherent Artin-Schelter regular Koszul dual \(\Lambda^!\). The BGG-correspondence is known to descend from the Koszul duality equivalence between bounded derived categories of graded modules over the two algebras, and one may ask whether something similar is true for the equivalence shown above.

We investigate this using a wider notion of Koszulity. GreenReitenSolberg'2002 present a notion of Koszulity for more general graded algebras, where the degree \(0\) part is allowed to be an arbitrary finite dimensional algebra. Madsen'2011 gives a simplified definition of \(T\)-Koszul algebras which is a generalization of the original one whenever the degree \(0\) part is of finite global dimension.

We generalize Madsen's definition to obtain the notion of \(n\)-\(T\)-*Koszul algebras*, where \(n\) is a positive integer and \(n=1\) returns Madsen's theory. We prove that an analogue of classical Koszul duality still holds in this generality, and we use this to give an answer to our motivating question.

Moreover, we provide the following characterization result: a finite dimensional graded Frobenius algebra of highest degree \(a \geq 1\) is \(n\)-\(T\)-Koszul if and only if \(\widetilde{T} = \oplus_{i=0}^{a-1}\Omega^{-ni}T \langle i \rangle\) is a tilting object in the associated stable category and the endomorphism algebra of this object is \((na-1)\)-representation infinite.

Furthermore, we show that when an \(n\)-representation infinite algebra has a right graded coherent higher preprojective algebra, then the bounded derived categories of graded modules over the trivial extension and over the associated \((n+1)\)-preprojective algebra are equivalent. Notice that in some sense, the theory we develop is a generalized Koszul dual version of parts of MinamotoMori'2011.

We also show that something similar holds in the higher representation finite case. Inspired by and seeking to generalize the notion of almost Koszul algebras as developed by BrennerButlerKing'2002, we arrive at the definition of \textit{almost \(n\)-\(T\)-Koszul algebras}. This enables us to show a similar characterization result as in the \(n\)-\(T\)-Koszul case."

"(topic withdrawn by author, will be automatically deleted in 24 hours unless flagged)"

Joint work with Lidia Angeleri Hügel (Università degli Studi di Verona)

Silting modules can be characterized as zero cohomologies of (not necessarily compact) two-term complexes. Ring epimorphisms with nice homological properties give rise to silting modules. It was shown in [Angeleri-Hügel,Marks,Vitória2016] that the universal localizations of a hereditary ring are parametrized by certain silting modules which are determined by a minimality condition and called minimal silting. A dual version of this result was recently established in [Angeleri-Hügel,Hrbek,arXiv] and led to the notion of minimal cosilting modules. The interest in minimal silting modules and minimal cosilting modules goes beyond the hereditary case. For example, every flat ring epimorphism starting in an arbitrary ring gives rise to a minimal cosilting module, and the converse holds true over any commutative noetherian ring.

In [Angeleri-Hügel,Sánchez2013] and [Buan,Krause2003], both large tilting and cotilting modules over \(R\), a tame hereditary algebra over a field \(k\), are parametrized by pairs \((Y,P)\), where \(Y\) is a branch module and \(P\) is a subset of the index set \(\mathbb{X}\) of the tubular family \(\mathbf{t}=\bigcup_{\lambda\in\mathbb{X}}\mathbf{t}_\lambda\) in the Auslander-Reiten quiver. We show that under the parametrization, minimal tilting or cotilting modules correspond to the pairs \((Y,P)\) where \(P\) is not empty. We construct a wide subcategory \(\mathcal{M}\) that allows to realize \(T_{(Y,P)}\) as a minimal tilting module arising from the universal localization \(R\rightarrow R_{\mathcal{M}}\) of \(R\) at \(\mathcal{M}\).

Theorem. If \(P\) is non-empty, then \(T_{(Y,P)}\) is equivalent to the tilting module \(T_{\mathcal{M}}=R_{\mathcal{M}}\oplus R_{\mathcal{M}}/R.\)

In [Buan,Krause2003] , large cotilting modules are determined by their indecomposable summands, which can be either finite dimensional regular modules, or infinite dimensional pure-injective modules, thus, Prüfer modules, adic modules, or the generic module.

Theorem. A large cotilting module is minimal if and only if it has an adic direct summand.

We know from the work of Breaz [Breaz2020] that every silting module \(T\) over a commutative ring \(R\) extends to a silting module \(T\otimes_{R}S \) along any ring epimorphism \(R\rightarrow S\). This can fail in the non-commutative case.

Example. Let \(R\) be the Kronecker algebra, and let \(T\) be a silting \(R\)-module. Then the \(R\)-module \(T\otimes_{R}S\) lies in \(\text{Gen}T\) for every homological ring epimorphism \(\lambda: R\rightarrow S\) if and only if \(T\) is not equivalent to the simple projective module \(P_{1}\).

In the commutative case, we see that minimality is often preserved by ring extensions.

Let \(R\) be a commutative noetherian ring, or a commutative ring of weak global dimension at most one. Then all minimal cosilting modules extend to minimal cosilting modules along any pseudoflat ring epimorphism.

Over a commutative hereditary ring, every  cosilting module  extends to a minimal cosilting module  along any ring epimorphism."

Mikhail Gorsky

*Joint work with Xin Fang.*

Hall algebras and various related structures play a prominent role in the modern representation theory. In their present form, they first appeared in a series of papers by Ringel on quantum groups. He introduced the notion of the Hall algebra of an abelian category with finite dimensional \(\mbox{Hom}\)- and \(\mbox{Ext}^1\)-spaces. As a vector space, it has a basis parameterized by the isomorphism classes of objects in the category. The multiplication captures information about the extensions between objects.

Ringel constructed an isomorphism between the twisted Hall algebra of the category of representations of a Dynkin quiver \(Q\) over the finite field \(\mathbb{F}_q\) and the nilpotent part of the corresponding quantum group, specialized at the square root of \(q.\) Later Green generalized this result to an arbitrary valued quiver \(Q\) by providing an isomorphism between the nilpotent part of the quantized universal enveloping algebra of the corresponding Kac-Moody algebra and the so-called *composition* (or *spherical*) subalgebra in \(\mathcal{H}_{tw}(\mbox{rep}_{\mathbb{F}_q}(Q))\) generated by the classes of simple objects. Using the Grothendieck group \(K_0(\mbox{mod}_{\mathbb{F}_q}(Q))\), he introduced an extended version of the Hall algebra which recovers the Borel part of the quantum group.

Hubery proved that the algebra defined in the same way as by Ringel, but for an exact category, is also unital and associative. Being abelian is a property of an additive category, while an exact category is an additive category endowed with an extra structure. The Hall algebra of an exact category depends not only on the underlying additive category, but on this structure as well. An additive category \(\mathcal{A}\) can be endowed with many different exact structures. They form a poset in a natural way, and BrüstleHassounLangfordRoy recently proved that it is a bounded complete lattice. Rump proved that each additive category \(\mathcal{A}\) admits a unique maximal exact structure \((\mathcal{A}, \mathcal{E}^{\mbox{max}}).\) The unique minimal exact structure is the split one.

To each exact category \(\mathcal{E}\), one can associate an abelian category of its *Auslander defects* \(\mathbf{def} (\mathcal{E}).\) By slightly modifying arguments of Enomoto and generalizing results of Buan, we obtain the following characterization of the lattice of exact structures.

Theorem 1. The lattice of exact structures on an arbitrary idempotent complete additive category \(\mathcal{A}\) is isomorphic to the lattice of Serre subcategories of the abelian category \(\mathbf{def} (\mathcal{E}^{\mbox{max}})).\)

Let \(\mathcal{A}\) be a \(\mbox{Hom}\)-finite \(\mathbb{F}_q\)-linear idempotent complete additive category. Let \(\mathcal{E}\) be an \(\mbox{Ext}^1\)-finite exact structure on \(\mathcal{A}.\) Consider a function \(w: \mbox{Iso}(\mathcal{A}) \to \mathbb{N}.\) We say that \(w\) is * *additive* if \(w(M \oplus N) = w(M) + w(N)\) for all \(M\) and \(N;\) * an \(\mathcal{E}\)-*quasi-valuation* if \(w(X) \leq w(M \oplus N)\) whenever there exists a conflation \(N \rightarrowtail X \twoheadrightarrow M\) in \(\mathcal{E};\) * an \(\mathcal{E}\)-*valuation* if it as an additive \(\mathcal{E}\)-quasi-valuation.

Such notions were motivated by Gröbner theory. For any object \(X\), \(w_X := \mbox{dim} \mbox{Hom} (X, -)\) is an additive function. If \(X\) is \(\mathcal{E}\)-projective, it is an \(\mathcal{E}\)-valuation. The function \(\mbox{dim} \mbox{End}(-)\) is a quasi-valuation for any exact structure on \(\mathcal{E},\) but it is usually not additive.

Theorem 2. Each \(\mathcal{E}\)-quasi-valuation \(w: \mbox{Iso}(\mathcal{A}) \to \mathbb{N}\) induces a filtration \(\mathcal{F}_w\) on \(\mathcal{H}(\mathcal{E}).\) If \(w\) is an \(\mathcal{E}\)-valuation, the associated graded algebra is the Hall algebra \(\mathcal{H}(\mathcal{E}')\) of an exact substructure \(\mathcal{E}' \leq \mathcal{E}\) on \(\mathcal{A}.\)

An additive category \(\mathcal{A}\) is *locally finite* if \(\forall X \in \mathcal{A},\) there exists only finitely many indecomposable \(Y, Z\) such that \(\mbox{Hom}(X, Y) \neq 0, \mbox{Hom}(Z, X) \neq 0.\)

Theorem 3. Suppose \(\mathcal{A}\) is locally finite. Then for each exact substructure \(\mathcal{E}' < \mathcal{E},\) there exists an \(\mathcal{E}\)-valuation \(w\) such that \(\mathbf{gr}_{\mathcal{F}_w}(\mathcal{H}(\mathcal{E})) = \mathcal{H}(\mathcal{E}').\) As \(w,\) one can take a (formal) sum of \(\mbox{dim}(\mbox{Hom}(X,-))\).

For the categories of representations of Dynkin quivers, we recover degenerations of the negative part of the corresponding quantum group. When applied to the maximal and to the minimal exact structures, Theorem 3 gives the quantum Poincare-Birkhoff-Witt theorem. Moreover, in this case (and, more generally, when \(\mathcal{A}\) has finitely many indecomposables), all \(\mathcal{E}^{\mbox{max}}\)-valuations form a simplicial cone. FangFourierReineke showed that it is related to Lusztig's negative tight monomial cone of a certain Lie algebra and, in type \(\tt A,\) can be identified with a maximal prime cone in the corresponding tropical flag variety.

In work in progress, we apply Theorem 2 and Theorem 3 to the category of 2-periodic complexes over \(\mbox{rep}_{\mathbf{F}_q}(Q)\) in order to realize all generalized quantum doubles of the Borel part of the corresponding quantum group. By works of Bridgeland and M.G., a certain quotient of a localized twisted Hall algebra of this category taken with the maximal (i.e. abelian) exact structure realizes the Drinfeld double of the quantum Borel subalgebra (Yanagida and LuPeng generalized this to the case of arbitrary \(\mbox{Hom}\)- and \(\mbox{Ext}^1\)-finite hereditary abelian categories and their twisted extended Hall algebras). We proved that the same construction for a certain smaller exact structure realizes the tensor double, which is the most degenerate generalized quantum double. We expect that all generalized quantum doubles can be realized via some of exact structures in the interval between these two in the lattice of exact structures on the category of 2-periodic complexes."

"<em>Joint work with Lleonard Rubio y Degrassi, based on arxiv.org/abs/2006.13871</em>

It is a central theme in representation theory to understand what structure possessed by a finite dimensional algebra is shared by algebras with equivalent derived categories or stable module categories. Stable equivalences are both more frequent and less understood than derived equivalences, making stable invariants an important tool. For any algebra \(A\), the Hochschild cohomology \(\mathrm{HH}^{\ast}(A)\) is a fundamentally important source of invariants. Its well-known Gerstenhaber algebra structure is enriched by a \(B_{\infty}\)-algebra structure on the Hoschschild cochain complex. It was shown in [Keller2003] that this \(B_{\infty}\)-structure is a derived invariant of \(A\).

Over a field of positive characteristic another operation on Hochschild cohomology arises from this \(B_\infty\)-structure, making \(\mathrm{HH}^{\ast}(A)\) into a restricted graded Lie algebra. The same graded Lie algebra can admit many different restricted structures, so it was asked by Linckelmann: <em> is the restricted Lie algebra structure of &nbsp;\({\rm HH}^{>0}(A)\) invariant under stable equivalences of Morita type for self-injective algebras? </em> A partial answer was given by the second author in [RubioyDegrassi2017] for the subclass of integrable derivations. We answer the question affirmatively:

<blockquote>

<strong>Theorem.</strong> Let \(A\) and \(B\) be finite dimensional, self-injective algebras over a field of positive characteristic. If \(A\) and \(B\) are stably equivalent of Morita type, then the induced transfer map on Hochschild cohomology gives an isomorphism \( {\rm HH}^{>0}(A)\cong {\rm HH}^{>0}(B)\) of (shifted) restricted graded Lie algebras.

As a consequence, all of the below (and more) are invariants under stable equivalences of Morita type:

* the restricted Lie algebra \({\rm HH}^1(A)\); * the induced restricted Lie algebra on the centre \({\sf Z}({\rm HH}^1(A))\) (which can be non-trivial); * the restricted universal enveloping algebras of \({\rm HH}^1(A)\) and of \({\sf Z}({\rm HH}^1(A))\); * the maximal rank of a \(p\)-toral subalgebra of \({\rm HH}^1(A)\). </blockquote>

The Theorem is a consequence of a more general result about the invariance of the \(B_\infty\)-structure under stable equivalences of Morita type, and so this is closely connected with a conjecture of Keller [Keller2018].

The main theorem in our paper applies more generally to a pair of Iwanaga Gorenstein algebras (with a correspondingly general notion of stable equivalence of Morita type). This is applied as an example to the Kn&ouml;rrer periodicity equivalence from commutative algebra.

Another application we give is to the classification of algebras of dihedral, semi-dihedral and quaternion type up to stable equivalence of Morita type [ZhouZimmermann2011] (the restricted structure fails to distinguish known classes, providing some evidence that in the open cases derived and stable equivalence classes should coincide).

We explain in detail how the restricted structure arises from the action of the \(B_\infty\)-operad, after the work in [Tourtchine2006]. Another operation on Hochschild cohomology arises in the same way (analogous to the Cohen-Dyer-Lashof operations from topology), and our arguments show the invariance of this as well. This operation does not seem to have been used in representation theory before."

Matthew Pressland (University of Leeds)

A dimer model \(D\) is traditionally a bipartite graph drawn on a torus, to which one can associate an (infinite-dimensional) associative algebra \(A_D\), the Jacobian algebra of the dual quiver with potential. Under various consistency conditions on \(D\), some combinatorial (or geometric) and some algebraic, Broomhead [Broomhead2012] showed that \(A_D\) is a (bimodule) \(3\)-Calabi–Yau algebra, and a non-commutative crepant resolution of its centre. Some of these results were later generalised to dimer models drawn on other surfaces.

There has recently been renewed interest in dimer models on surfaces with boundary, obtained from those on closed surfaces by cutting out discs, avoiding the vertices of the bipartite graph. Thus one obtains a bipartite graph in the resulting surface with boundary, together with 'half-edges', connecting a vertex to the boundary of the surface rather than to another vertex. For such a dimer model \(D\), one can attach an algebra \(A_D\), an example of a frozen Jacobian algebra.

The best understood case is that of dimer models in discs. Here, the analogue of Broomhead's geometric consistency condition, which places restrictions on the way in which certain zig-zag paths in the disc determined by \(D\) may intersect, is that these paths form a Postnikov diagram [Postnikov2006], a combinatorial object familiar from Postnikov's study of the positroid stratification of the Grassmannian, and the definition of cluster algebra structures on the coordinate rings of the resulting (open) positroid varieties [GalashinLam2019].

!Screen Shot 2020-10-27 at 16.59.32|690x225, 100%

The above figure shows a dimer model in the disc (left), its collection of zig-zag paths (centre) and the quiver of the algebra \(A_D\) (right). Note that a pair of zig-zag paths meeting twice are oriented in opposite directions between these crossings—this is the most important part of the definition of a Postnikov diagram. The quiver naturally has faces, certain distinguished cycles, because of its embedding into the disc, and each arrow belongs to either one or two faces; those in two faces are shown in green in the figure. The algebra \(A_D\) is the complete path algebra of the quiver modulo the relations \(p=q\) whenever \(p\) and \(q\) are paths such that \(pa\) and \(qa\) are the two faces containing an arrow \(a\).

In [Pressland2019] we prove a generalisation of Broomhead's theorem for connected consistent dimer models in the disc. For such a dimer model \(D\), let \(e\in A_D\) be the idempotent given by summing the vertex idempotents at the boundary vertices of the quiver (shown in blue in the figure). Then \(A_D\) is internally bimodule \(3\)-Calabi–Yau with respect to \(e\), in the sense of [Pressland2017]. For example, this means that the \(3\)-Calabi–Yau duality formula \[\operatorname{Ext}^i_{A_D}(M,N)=\operatorname{Ext}^{3-i}_{A_D}(N,M)^*\] holds whenever \(N\) is a finite-dimensional module with \(eN=0\), i.e. when \(N\) is supported at the internal (green) vertices of the quiver. Moreover, \(A_D\) has global dimension at most \(3\) (and precisely \(3\) as long as there is at least one internal vertex).

As well as extending Broomhead's result beyond closed surfaces, this result has consequences for the categorification of cluster algebra structures on positroid varieties. For \(e\) as in the preceding paragraph, define \(B=eA_De\) and \(T=eA_D\). Then \(B\) is an Iwanaga–Gorenstein algebra such that the Frobenius category \[\operatorname{GP}(B)=\{X\in\operatorname{mod}{B}:\text{\(\operatorname{Ext}^i_B(X,B)=0\) for all \(i>0\)}\}\] of Gorenstein projective \(B\)-modules is stably \(2\)-Calabi–Yau, and \(T\in\operatorname{GP}(B)\) is a cluster-tilting object with endomorphism algebra isomorphic to \(A_D\). These properties make \(\operatorname{GP}(B)\) an ideal candidate for a categorification of the cluster algebra structure on the positroid variety associated to \(D\). Indeed, for very special dimer models \(D\), whose corresponding positroid is open in the Grassmannian, the category \(\operatorname{GP}(B)\) is equivalent to Grassmannian cluster category of Jensen, King and Su [JensenKingSu2016] categorifying this cluster algebra structure. In these cases, the isomorphism of \(A_D\) with the endomorphism algebra of \(T\) is due to Baur, King and Marsh [BaurKingMarsh2016].

Making the connection between \(\operatorname{GP}(B)\) and the corresponding positroid cluster algebra more precise, for example by giving a full explanation of the cluster character, is the subject of ongoing work, some joint with Çanakçı and King. With these collaborators, we show (in a paper to appear) that each of the categories \(\operatorname{GP}(B)\) obtained from a consistent dimer model as above appears as a full subcategory of Jensen–King–Su's Grassmannian cluster category."

Marco Armenta (Université de Sherbrooke)

Joint work with Pierre-Marc Jodoin (Université de Sherbrooke)[ArmentaJodoin2020]

The greatest advances in artificial intelligence are a consequence of the use of neural networks. Despite years of research, a mathematical understanding of the performance and behaviour of neural networks has been elusive. Even more, each mathematical analysis of neural networks makes strong assumptions on the combinatorics of the network and the data it processes.

We show that quiver representation theory provides a language for neural networks that do not require any assumptions to be made on the combinatorics of the network nor the data it processes. Moreover, the use of this language preserves commonly used tools to study neural networks and generalizes known properties.

We work over the field of complex numbers \(\mathbb{C}\). A network quiver \(Q\) is a finite quiver that has no oriented cycles other than loops on vertices that are not source nor sink vertices, and there is only one loop for each of those vertices. Let \(d\) be the number of source vertices of \(Q\) and \(K\) the number of sinks. The delooped quiver of \(Q\) is the quiver \(Q^\circ\) obtained by removing all loops from \(Q\), and the hidden quiver \(\widetilde{Q}\) is the full subquiver of the delooped quiver obtained by removing all source and sink vertices.

A neural network over a network quiver \(Q\) is a pair \((W,f)\) where \(W\) is a thin representation of the delooped quiver of \(Q\) and \(f\) are activation functions (non-linear) assigned to the loops of the network quiver. We will consider \(W\) as a stable double-framed thin quiver representation in the sense of Def 2.1 of [Reineke2008] together with its dual notion. The group action of the change of basis group of double-framed thin representations \(G:= \prod_{v \in \widetilde{Q}_0} \mathbb{C}^*\) extends to the activation functions by \((\tau_v \cdot f_v)(x) = \tau_v f(x/\tau_v)\) for each \(v \in \widetilde{Q}\), and then we have an action of the group \(G\) on neural networks over \(Q\).

Neural networks are use to compute in the following way. Given an element \(x\in \mathbb{C}^d\) that we interpret as a vector indexed by the source vertices of \(Q\), the activation output of a vertex \(v\in Q_0\) is defined as \(a(W,f)_v(x)=x_v\) if \(v\) is a source vertex and \(a(W,f)_v(x) = f_v \left( \sum_{\alpha \in v^-} W_\alpha a(W,f)_{s(\alpha)}(x) \right)\) in any other case. The network function of a neural network \(\Psi(W,f): \mathbb{C}^d \to \mathbb{C}^K\) takes \(x\in \mathbb{C}^d\) to the vector \(\Psi(W,f)(x) \in \mathbb{C}^K\) given by the activation output of the sink vertices of \(Q\).

Theorem: The network function is invariant under the action of \(G\).

A thin representation \(V\) of the delooped quiver \(\widetilde{Q}\) induces a neural network by using identity activation functions, that we denote \((V,1)\). Given a neural network \((W,f)\) and a vector \(x \in \mathbb{C}^d\) we construct a stable double-framed thin quiver representation \(W_x^f\) with the following property.

Theorem: Let \(Q\) be a network quiver, \((W,f)\) a neural network over \(Q\) and \(x\in \mathbb{C}^d\), and denote \(1^d := (1,...,1) \in \mathbb{C}^d\). Then \(\Psi(W,f)(x) = \Psi(W_x^f,1)(1^d)\).

What this theorem means is that the neural network *understands* the data as the quiver representations \(W_x^f\). Moreover, these are invariant under the action of \(G\), and therefore the isomorphism classes of the data-representations \([W_x^f]\) lie in the moduli space \(_d\mathcal{M}_K(\widetilde{Q})\) of stable double-framed thin quiver representations, which is constructed by GIT. We denote by \(\varphi(W,f) : \mathbb{C}^d \to _d\mathcal{M}_K(\widetilde{Q})\) the map \(x \mapsto [W_x^f]\), and we define \(\hat{\Psi} : _d\mathcal{M}_K(\widetilde{Q}) \to \mathbb{C}^K\) by \(\hat{\Psi}([V]) := \Psi(V,1)(1^d)\). Let \(NN(Q)\) be the set of isomorphism classes of neural networks over \(Q\) under the action of \(G\). Finally, we have the following result.

Theorem: Let \(Q\) be a network quiver. There is a commutative diagram !diagram|447x134, 100% What this theorem means for neural networks is that the learning process (given by an optimization algorithm) of a neural network reshapes the submanifold \(Im(\varphi(W,f))\) inside the moduli space \(_d\mathcal{M}_K(\widetilde{Q})\)."

Xiaofa Chen (USTC) Xiao-Wu Chen (USTC)

Let \(k\) be a field and \(A\), \(B\) be finite dimensional \(k\)-algebras. It is a well-known conjecture posed by Rickard whether any triangle equivalence between bounded derived categories \(\mathbf D^b(A)\) and \(\mathbf D^b(B)\) is standard, that is, isomorphic to the derived tensor functor \(X\otimes^{\mathbb L}_{A} -\) by a two-sided tilting complex \(_{B}X_{A}\).

This question has been confirmed in several cases, such as hereditary algebras, Fana algebras, triangular algebras, the algebra of dual numbers and derived-discrete algebras of nite global dimension. For the notion unexplained, we refer to

Proposition There is a triangle equivalence $\mathrm{rep}(\mathbf D^b _{dg}(A{-}\mathrm{mod}),\mathbf D^b_{ dg}(B{-}\mathrm{mod})) \rightarrow \{M\in \mathbf D(B\otimes A^{op})|M_A \ \mathrm{is} \ \mathrm{perfect}\}$ sending a quasi-functor \(X\) to \(X(B,A)\). As a consequence, a triangle functor \(F: \mathbf D^b(A{-}\mathrm{mod}) \rightarrow \mathbf D^b(B{-}\mathrm{mod})\) is liftable if and only if it is standard.

For an abelian category \(\mathcal C\), a triangle functor \(F:\mathbf D^b(\mathcal C)\rightarrow \mathbf D^b(\mathcal C)\) is called pseudo-identity if \(F(X)=X\) for each complex \(X\) and \(F|_{\Sigma^n\mathcal C}:\Sigma^n\mathcal C\rightarrow \Sigma^n\mathcal C\) is the identity functor for each \(n\). Proposition Let \(F:\mathbf D^b(A{-}\mathrm{mod}) \rightarrow \mathbf D^b(\mathcal C)\) be a triangle equivalence. Then there is a factorization \(F\simeq F_2 \circ F_1\) of triangle functors, where \(F_1:\mathbf D^b(A{-}\mathrm{mod}) \rightarrow \mathbf D^b(A{-}\mathrm{mod})\) is a pseudo-identity and \(F_2:\mathbf D^b(A{-}\mathrm{mod}) \rightarrow \mathbf D^b(\mathcal C)\) is a liftable equivalence.

For a triangle functor \(F\) between bounded derived categories of coherent sheaves over smooth projective schemes, it is known by Toën and Lunts and Schnürer that \(F\) is of Fourier-Mukai type if and only if it is liftable. So we get our desired result

Theorem Let \(A\) and \(B\) be finite dimensional algebras. Assume there is a triangle equivalence between \(\mathbf D^b(A{-}\mathrm{mod})\) and \(\mathbf D^b(\mathrm{coh-}\mathbb X)\) with \(\mathbb X\) a smooth projective scheme. Then any triangle equivalence \(F:\mathbf D^b(A{-}\mathrm{mod})\rightarrow \mathbf D^b(B{-}\mathrm{mod})\) is standard."

Jordan McMahon (unaffiliated)

In geometry and combinatorics, a common object of study is that of an *arrangement* - for example arrangements of lines or of hyperplanes. Within this class, an *arrangement of great (pseudo)circles on the sphere* consists of a family \(\{c_1,...,c_n\}\) of simple closed curves, called (pseudo)circles, such that * Every two circles have exactly two points in common at which they cross. * Crossings are simple (only two circles may cross at one time) and transverse (when circles meet, they must cross). * The intersection of any two circles \(c_i\) and \(c_j\) is (uniquely) transversed by all remaining circles.

Note that in other contexts, a different notion might be used. Great circle arrangements on the sphere are commonly drawn in two dimensions. !figure1|501x357, 75%

A classic result is that there is a correspondence between great circle arrangements and the class of wiring diagrams with reversed endpoints. !figure2|655x315, 75%

The correspondence between wiring diagrams and great circle arrangements is not bijective, since there are two possible wiring diagrams with three wires and reversed endpoints, whereas there is only one great circle arrangement having three circles.

On the other hand, cluster algebras arising from the Grassmanian are a current trend in representation theory. They are commonly depicted by *\((k,n)\)-alternating strand diagrams*, following Scott03. The case \(n=2k\) is especially of interest, since \((k,2k)\)-alternating strand diagrams reduce to the *double wiring diagrams* of FominZelevinsky99. !figure3|630x261, 75%

Note that we may simply twist the strands so that the double wiring arrangement has endpoints that become reversed. Hence \((k,2k)\)-alternating strand diagrams may also be described by great circle arrangements.

To generalise this, define a *\((k,n)\)-arrangement of circles on the sphere* to consist of a family \(\{c_1,\ldots ,c_n\}\) of circles, such that * For each circle and for at least \(2k-1\) of the remaining circles, there are exactly two points in common at which they cross. * Crossings are simple and transverse. * The intersection of any two circles \(c_i\) and \(c_j\) is uniquely transversed by precisely \(2k-2\) of the remaining circles.

Note that we no longer refer to great circles, since the term "great" indicates that all circles intersect. In case \(n=2k\), then this definition restricts to a great circle arrangement.!figure4|690x427, 75%

Conjecture: There is a correspondence between \((k,n)\)-alternating strand diagrams and \((k,n)\)-arrangements of circles on the sphere.

Note that a correspondence would not be bijective, since mutations may be performed on one hemisphere without affecting the other. From a representation theory standpoint, such a description would simplify the description of the associated dimer algebras. Typically, dimer algebras that arise from alternating strand diagrams are defined using a quiver with potential, whilst ignoring boundary relations. By describing first a dimer algebra on the sphere, we remove this potential complication."

Souheila Hassoun (Université de Sherbrooke)

Joint work with Thomas Brüstle and Aran Tattar

In 1889, Otto Hölder reinforced Camille Jordan’s theorem by proving what is known as the Jordan-Hölder-Schreier theorem, which states that any two composition series of a given group are equivalent, that is, they have the same length and the same factors, up to permutation and isomorphism. Most proofs use the concept of intersection and sum, which is readily available for objects in an abelian category. We generalise this work to the setup of Quillen exact categories Quillen1973.

We say an exact category \((\mathcal{A},\mathcal{E})\) is a Jordan-Hölder exact category if any two finite \(\mathcal{E} \)-composition series of \(X\) are equivalent. This is an interesting problem since the Jordan-Hölder property does not hold in general for any exact category, see counter-example 6.9 of BrustleHassounLangfordRoy2018. This problem is also studied by Enomoto in Enomoto2019, using the Grothendieck monoid.

In Baumslag2006 a short proof of the Jordan-Hölder theorem for groups is given by intersecting the terms of one subnormal series with those in another series. Motivated by these ideas, we generalise the abelian notions of intersection and sum to exact categories. We do this in two ways. Firstly, by considering intersections as pullbacks and sums as pushouts, we recall Admissible Intersection (AI) and Admissible Intersection and Sum (AIS) categories from HassounRoy2019. The AI-categories are pre-abelian exact categories where admissible monics are stable under pullback along admissible monics and all such pullbacks exist. We prove that the AI-categories are necessarily quasi-abelian with the maximal exact structure. It has been proved recently in HassounShahWegner2020, that the converse is also true. Hence, we obtain a new characterisation of quasi-abelian categories.

The AIS-categories are the AI-categories that satisfy the additional property that the unique induced morphism in the pushout coming from the pullback diagram is an admissible monic. It turns out that the AIS-categories are precisely the abelian categories endowed with the maximal exact structure. This, along with our study of the behaviour of admissible morphisms under composition and sum, allows us to give many alternative characterisations of abelian categories. We also establish a useful generalisation of the Fourth Isomorphism Theorem for modules.

We observe that the pullback and pushout notions of unique intersection and sum do not necessarily apply to general exact categories - even if the exact category has the Jordan-Hölder property. This leads us to define a general notion of admissible intersection, sum and \(\mathcal{E}\)-Jacobson radical that works for all exact categories. We then use this to introduce the \(\mathcal{E}\)-Artin-Wedderburn categories, and show that any Krull-Schmidt \(\mathcal{E}\) -Artin-Wedderburn category \((\mathcal{A}, \mathcal{E})\) is a Jordan-Hölder exact category.

We finally give for any Nakayama algebra an explicit description of all exact structures and characterise the Artin-Wedderburn exact structures among them:

Let \(\mathcal{A }= mod \Lambda\) where \(\Lambda\) is a Nakayama algebra. Then the exact category \((\mathcal{A},\mathcal{E})\) is \(\mathcal{E}\)-Artin-Wedderburn precisely when it is Jordan-Hölder."

Wassilij Gnedin (Ruhr University Bochum)

The guiding goal of this study is to compare the derived representation theory of a ring \(\Lambda\) to that of some quotient \(\overline{\Lambda}\) using the push-down functor \(\) \mathbf{F}\!: \mathsf{D}^{-}(\mathsf{mod}\, \Lambda) \longrightarrow \mathsf{D}^-(\mathsf{mod} \,\overline{\Lambda}), \qquad L^{\bullet}\, \longmapsto L^{\bullet} {\overset{\mathbb{L}}{\otimes}}_{\Lambda}\, \overline{\Lambda}. \(\) Unfortunately, the functor \(\mathbf{F}\) is usually neither injective nor surjective on isomorphism classes of objects. The problem whether a given complex \(P^{\bullet} \in \mathsf{D}^-(\mathsf{mod}\, {\overline{\Lambda}})\) admits some preimage \(L^{\bullet} \in \mathsf{D}^{-}(\mathsf{mod}\,\Lambda)\) under the functor \(\mathbf{F}\) has been studied in various setups by Green [Green1959] , Eisenbud [Eisenbud1980], Rickard [Rickard1991], Auslander, Ding and Solberg [AuslanderDingSolberg1993], and Yoshino [Yoshino1997].

To obtain our main result we extend Rickard's lifting techniques [Rickard1991] to a common denominator of these setups, which includes the case where

* \(\Lambda\) is an \(\mathsf{R}\)-algebra over a commutative complete local ring \(\mathsf{R}\) such that \(\Lambda\) is finitely generated as \(\mathsf{R}\)-module, and

* \({\overline{\Lambda}}\) is the quotient \(\Lambda/\mathbf{x}\Lambda\) by a non-zero divisor \(\mathbf{x}\) from the maximal ideal \(\mathfrak{m}\) of \(\mathsf{R}\) which acts also as a non-zero divisor on \(\Lambda.\)

In particular, \(\mathbf{x}\) commutes with any element of \(\Lambda\).

The focus on the particular type of objects in the next result is motivated by work of Aihara and Iyama [AiharaIyama2012].

In the setup above, the push-down functor \(\mathbf{F}\) induces an isomorphism of posets of silting complexes
\(\)
\psi_s:

(\mathsf{silt}\, \Lambda,\leq) \overset{1:1}{\longrightarrow} (\mathsf{silt}\, {\overline{\Lambda}}, \leq).

\(\)
Furthermore, the map \(\psi_s\) restricts to a bijection between tilting complexes of \(\Lambda\) with \(\mathbf{x}\)-regular endomorphism ring and tilting complexes of \({\overline{\Lambda}}\).

Varying the regular element \(\mathbf{x}\) yields families of rings with the same silting theory. For instance, the arrow ideal completion \(\Gamma\) of the preprojective algebra of affine type \(\widetilde{\mathbb{A}}\) is an *order* over the power series ring \(\mathsf{R} = \mathbf{k} [| s,t |]\), that is, an \(\mathsf{R}\)-algebra with \(\Lambda_{\mathsf{R}} \in \mathsf{add}\,\mathsf{R}.\) For almost any choice \(\mathbf{x}\) from the maximal ideal \(\mathfrak{m} = (s,t)\) of the ring \(\mathsf{R}\) the quotient \(\Gamma/\mathbf{x}\Gamma\) as well as the order \(\Gamma\) are derived-wild. It turns out that there is a special choice \(\mathbf{x}_0 \in \mathfrak{m}\) such that \(\Gamma/ {\mathbf{x}}_0 \Gamma\) is a *ribbon graph order*, that is, the arrow ideal completion of the path algebra of a complete gentle quiver. Because the \(\mathsf{R}\)-order \(\Gamma\) is symmetric, the above result yields bijections on tilting complexes \[\mathsf{tilt}\, \Gamma/{\mathbf{x}}_0 \Gamma\overset{1:1}{\longleftarrow} \mathsf{tilt}\, \Gamma\overset{1:1}{\longrightarrow} \mathsf{tilt}\, \Gamma/\mathbf{x}\Gamma\] In this example, the classification problem of tilting complexes of a family of derived-wild rings can be reduced to the single derived-tame case of \(\Gamma/{\mathbf{x}}_0 \Gamma\).

Further properties of the push-down functor yield the following consequence:

If \(\mathcal{C}\) is a class of Noetherian \(\mathbf{x}\)-regular \(\mathsf{R}\)-algebras which is closed under derived equivalences, then the class \(\overline{\mathcal{C}}\) of their quotients by the regular element \(\mathbf{x}\) is closed under derived equivalences as well.

More specifically, this statement can be applied to ribbon graph orders to show that *twisted Brauer graph algebras* defined in [Gnedin2019] are closed under derived equivalences.

At last, we consider a setup where the element \(\mathbf{x}\) does not have to commute with any element of the Noetherian \(\mathsf{R}\)-algebra \(\Lambda\). More precisely, we consider quotients

* \(\overline{\Lambda}\) of the form \(\Lambda/\mathbf{x}\Lambda,\) where \(\mathbf{x}\) is a non-zero divisor from the Jacobson radical \(\mathrm{rad}\,\Lambda\) such that there is an automorphism \(\nu \in \mathrm{Aut}_{\mathsf{R}}\, \Lambda\) with \(\mathbf{x} \, a = \nu(a) \, \mathbf{x}\) for any element \(a \in \Lambda.\)

The bijection of the previous setup allows for a modification to \(\nu\)-invariant silting complexes in the new setup. The upshot is that it may happen that *any silting complex of \(\Lambda\) is \(\nu\)-invariant, although the automorphism \(\nu\) is not trivial*. This phenomenon occurs in connection with ribbon graph orders and Brauer graph algebras."

Sergio Pavon (University of Padova)

The content of this research snapshot is Section 5 of a joint work with J. Vitória, "Hearts for commutative noetherian rings: torsion pairs and derived equivalences", available online at https://arxiv.org/abs/2009.08763 .

Let \(\mathcal{A},\mathcal{B}\) be two Grothendieck categories. An interesting problem is to decide whether they are derived equivalent, i.e. if there is a triangle equivalence \(\mathsf{D}(\mathcal{A})\to \mathsf{D}(\mathcal{B})\) between their derived categories (or the bounded versions). This problem becomes easier to takle when \(\mathcal{B}\) is the heart of an intermediate (i.e., with heart concentrated in finitely many degrees) \(t\)-structure \(\mathbb{T}\) in \(\mathsf{D}(\mathcal{A})\): this is for example the setting in co/silting theory, when \(\mathbb{T}\) is determined by a co/silting object of \(\mathsf{D}^b(\mathcal{A})\) (see e.g [MarksVitória2018], [AngeleriHügelMarksVitória2016] and references therein). In this situation, a triangle functor \(\mathsf{D}^b(\mathcal{B})\to \mathsf{D}^b(\mathcal{A})\), called the bounded realisation functor of (the heart of) \(\mathbb{T}\), becomes available ([BeilinsonBernsteinDeligne1982]), leaving the question of whether it is an equivalence (in the co/silting case, see [PsaroudakisVitória2018]).

Aside from tilting theory, and partly in a generalisation of it, another source of intermediate \(t\)-structures in \(\mathsf{D}(\mathcal{A})\) was introduced in [HappelReitenSmalø1996]. Starting from a torsion pair \(\mathbf{t}=(\mathcal{T},\mathcal{F})\) in \(\mathcal{A}\), the authors constructed a \(t\)-structure \(\mathbb{T}_\mathbf{t}\) in \(\mathsf{D}(\mathcal{A})\), having heart \(\mathcal{H}_\mathbf{t}:=\mathcal{F}[1]\ast\mathcal{T}\). That is, \(\mathcal{H}_\mathbf{t}\) consists of complexes whose cohomologies are torsionfree in degree \(-1\), torsion in degree \(0\), and zero elsewhere.

Addressing the question of whether \(\mathcal{H}_\mathbf{t}\) is derived equivalent to \(\mathcal{A}\), in [ChenHanZhou2019] the authors proved a very nice criterion: the bounded realisation functor \(\mathsf{D}^b(\mathcal{H}_\mathbf{t})\to \mathsf{D}^b(\mathcal{A})\) is an equivalence if and only if every object \(X\) of \(\mathcal{A}\) lies in a sequence \[0\to F_0\to F_1\to X\to T_0\to T_1\to 0\] such that * \(F_0,F_1\in\mathcal{F}\) and \(\mathcal{T}_0,\mathcal{T}_1\in\mathcal{T}\) * it represents the zero element of \(\operatorname{Ext}_\mathcal{A}^3(T_1,F_0)\).

We will call such a sequence a CHZ-sequence. In [ChenHanZhou2019] various applications of this criterion are explored, retrieving some known results and obtaining some new ones. In Section 5 of the paper presented in this research snapshot, the criterion is applied to the specific case of a hereditary torsion pair \(\mathbf{t}\) in the Grothendieck category \(\mathcal{A}\), leading to some very precise conclusions. A first observation, holding for any torsion pair in \(\mathcal{A}\), is that a CHZ-sequence exists for every object \(X\) of \(\mathcal{A}\) if and only if it exists for a generator \(G\) of \(\mathcal{A}\).

If we focus on hereditary torsion pairs, the condition further simplifies: if \(\mathbf{t}=(\mathcal{T},\mathcal{F})\) is a hereditary torsion pair in \(\mathcal{A}\), then a CHZ-sequence exists for an object \(X\) of \(\mathcal{A}\) if and only if there is a sequence \[F\to X\to T\to 0 \qquad \text{ with } F\in\mathcal{F} \text{ and } T\in\mathcal{T}.\] Such a sequence will be called a short CHZ-sequence. This fact, which has a very simple proof, allows one to get rid of the difficult part of the criterion by [ChenHanZhou2019], which is the vanishing of the 3-extension.

As a last step, we try to identify, among all morphisms \(\operatorname{Hom}(\mathcal{F},G)\), the best candidate to produce a short CHZ-sequence for a generator \(G\) of \(\mathcal{A}\): this leads to the following result. Let \(\mathbf{t}\) be a hereditary torsion pair in \(\mathcal{A}\), with torsion radical \(t\). Then a short CHZ-sequence exists for a generator \(G\) of \(\mathcal{A}\) if and only if the sequence \[G/t(G)^{(\operatorname{Hom}_{\mathcal{A}}(G/t(G),G))}\overset{\pi}{\to} G \to G/\operatorname{tr}_{G/t(G)}(G) \to 0\] has \(G/\operatorname{tr}_{G/t(G)}(G)\) in \(\mathcal{T}\) (and so it is itself a short CHZ-sequence). Here \(\operatorname{tr}_{G/t(G)}(G)\) denotes the trace of \(G/t(G)\) in \(G\), and it is defined as the cokernel of the canonical morphism \(\pi\).

Combining all the previous results, we get:

Theorem 1. Let \(\mathcal{A}\) be a Grothendieck category, \(\mathbf{t}\) be a hereditary torsion pair in \(\mathcal{A}\) with torsion radical \(t\), and \(G\) a generator of \(\mathcal{A}\). Then the bounded realisation functor \(\mathsf{D}^b(\mathcal{H}_\mathbf{t})\to \mathsf{D}^b(\mathcal{A})\) is an equivalence if and only if \(G/\operatorname{tr}_{G/t(G)}(G)\) is torsion.

To conclude this snapshot, we provide an interesting application of this new criterion.

Theorem 2. Let \(R\) be a commutative noetherian ring. Then every hereditary torsion pair in \(\operatorname{Mod}(R)\) gives rise to a bounded derived equivalence, as above. *Proof.* One checks that \(\operatorname{tr}_{R/t(R)}(R)=\operatorname{Ann}_R(t(R))\), the annihilator of the torsion ideal. Now it is easy to see that, if \(t(R)\) is generated by \(x_1,\dots,x_n\), the module \(R/\operatorname{Ann}(t(R))\) embeds into the finite product \(t(R)^{\oplus n}\) via the map induced by \(1\mapsto (x_1,\dots, x_n)\). Now, this finite product is a torsion module, and the torsion pair is hereditary, so \(R/\operatorname{tr}_{R/t(R)}(R)\) is torsion.

As a last remark, in this commutative noetherian case, which is the main subject of the paper, two phenomena occur. First, this bounded derived equivalence automatically extends to an unbounded one. Second, the \(t\)-structures obtained from hereditary torsion pairs in \(\operatorname{Mod}(R)\) are precisely those induced by 2-term cosilting complexes: therefore, the theorem above could be rewritten as saying that every 2-term cosilting complex in \(\mathsf{D}(R)\) is cotilting."

Pierre-Guy Plamondon (Université de Versailles Saint-Quentin)

*Joint work with Toshiya Yurikusa (Tohoku University).*

The \(\mathbf{g}\)-vector fan of an Artin algebra \(\Lambda\) is a polytopal, simplicial fan that encodes information about \(\tau\)-rigid modules, or equivalently about \(2\)-term presilting complexes of projective \(\Lambda\)-modules.

Setup. Let \(\Lambda\) be a basic Artin algebra, and let \(\Lambda = \bigoplus_{i=1}^n P_i\) be its decomposition into indecomposable projective modules. Let \(K^{[-1,0]}(\operatorname{proj}(\Lambda))\) be the category of *\(2\)-term complexes* of finitely generated projective modules, that is, complexes concentrated in cohomological degrees \(-1\) and \(0\). Denote by \(M\) its Grothendieck group, and by \(M_{\mathbb{R}}\) the extension of scalars \(M \otimes_{\mathbb{Z}} \mathbb{R}\).

The *\(\mathbf{g}\)-vector* of an object \(X\) of \(K^{[-1,0]}(\operatorname{proj}(\Lambda))\) is its class in the Grothendieck group \(M\).

It follows from results of [DehyKeller2008], [AdachiIyamaReiten2014] and [DerksenFei2015] that the \(\mathbf{g}\)-vectors of rigid objects form a *polyhedral, simplicial fan* \(\mathcal{F}_\Lambda\) in \(M_{\mathbb{R}}\) whose rays are generated by the \(\mathbf{g}\)-vectors of indecomposable rigid objects, and whose cones are generated by those of pairwise \(\operatorname{Ext}^1\)-orthogonal indecomposable rigid objects.

Definition. The *\(\mathbf{g}\)-vector fan* of \(\Lambda\) is the fan \(\mathcal{F}_\Lambda\) in \(M_\mathbb{R}\).

Although they are defined in terms of \(\tau\)-tilting theory, \(\mathbf{g}\)-vector fans are related to stability conditions, and have applications to \(\mathbf{g}\)-vector fans of cluster algebras. They are also related to the scattering diagrams of [GrossHackingKeelKontsevich2018].

If \(\Lambda\) is \(\tau\)-tilting finite over a field, then its \(\mathbf{g}\)-vector fan is known to cover the whole of \(M_\mathbb{R}\) [DemonetIyamaJasso2019]. Our main result deals with algebras whose fan is not complete, but dense in \(M_\mathbb{R}\). Such algebras are said to be *\(\mathbf{g}\)-tame*.

Theorem. Let \(\Lambda\) be a finite-dimensional basic algebra over an algebraically closed field. If \(\Lambda\) is tame, then it is \(\mathbf{g}\)-tame.

The theorem generalizes known results for extended Dynkin quivers [Hille2006], Jacobian algebras from triangulated surface [Yurikusa2020], gentle algebras and special biserial algebras [AokiYurikusa2020+] AsaiDemonetIyama] (private communication). Its proof uses the generic decomposition of \(\mathbf{g}\)-vectors [[P.2013], and a more precise statement of it for tame algebras [GeissLabardini-FragosoSchröer2020+]. It uses a variation of twist functors to approach any rational ray in the boundary of \(\mathcal{F}_\Lambda\) by \(\mathbf{g}\)-vectors of rigid objects. The functors used were studied as *truncated twist functors* in an appendix by B.Keller to [P.-Yurikusa2020+].

As a corollary, using results of [GeissLabardini-FragosoSchröer2016] on tame Jacobian algebras, we get a near classification of the quivers for which the corresponding cluster algebra has a dense \(\mathbf{g}\)-vector fan.

Corollary. Let \(Q\) be a connected quiver with at least three vertices and without loops or \(2\)-cycles. Then the \(\mathbf{g}\)-vector fan of the corresponding cluster algebra is dense or its closure is a closed half space if and only if \(Q\) is mutation finite, *except possibly if \(Q\) is of type \(X_6\) or \(X_7\)*.

Conjecture. The \(\mathbf{g}\)-vector fan in type \(X_6\) is dense, and the closure of that in type \(X_7\) is a half-space."

Pérez, Marco A. (Instituto de Matemática y Estadística. Universidad de la República)

(Joint work with Huerta, Mindy and Mendoza, Octavio).

We present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations restricted to certain subcategories. Moreover, there are some connections between cut cotorsion pairs and Auslander-Buchweitz approximation theory, by considering relative analogs for Frobenius pairs (in the sense of [BecerrilMendozaPérezSantiago2019]) and Auslander-Buchweitz contexts. Some applications of this notion can be given in the settings of relative Gorenstein homological algebra, chain complexes and quasi-coherent sheaves, as well as to characterize some important results on the *Finitistic Dimension Conjecture*, the existence of right adjoints of quotient functors by Serre subcategories, and the description of cotorsion pairs in triangulated categories as co-\(t\)-structures.

Definition: Given an abelian category \(\mathcal{C}\) and a (full) subcategory \(\mathcal{S} \subseteq \mathcal{C}\), we say that two classes \(\mathcal{A}\) and \(\mathcal{B}\) of objects of \(\mathcal{C}\) form a (complete) left cotorsion pair cut along \(\mathcal{S}\) if: (1) \(\mathcal{A}\) is closed under direct summands; (2) \(\mathcal{A} \cap \mathcal{S} = {}^{\perp_1}\mathcal{B} \cap \mathcal{S} = \{ M \in \mathcal{C} : \operatorname{Ext}^1_{\mathcal{C}}(M,B) = 0, \text{ } \forall B \in \mathcal{B} \} \cap \mathcal{S}\); and (3) if every object of \(\mathcal{S}\) is the epimorphic image of an object in \(\mathcal{A}\) with kernel in \(\mathcal{B}\). Dually, one has the concept of right cotorsion pair cut along \(\mathcal{S}\). A pair \((\mathcal{A,B})\) is a cotorsion pair cut along \(\mathcal{S}\) if it is both a left and right cotorsion pair cut along \(\mathcal{S}\).

One basic example of a cut cotorsion pair can be found in the category \(\mathsf{Mod}(R)\) of (left) \(R\)-modules over an arbitrary ring \(R\). The classes \(\mathcal{GP}(R)\) of Gorenstein projective modules and \(\mathcal{P}(R)^\wedge\) of modules with finite projective dimension form a cotorsion pair \((\mathcal{GP}(R),\mathcal{P}(R)^\wedge)\) cut along \(\mathcal{GP}(R)^\wedge\), the subcategory of modules with finite Gorenstein projective dimension. This carries over to more general contexts in relative Gorenstein homological algebra, such as Ding projective modules [Gillespie2010], AC-Gorenstein projective modules [BravoGillespieHovey2014], and Gorenstein objects in abelian categories relative to GP-admissible pairs [BecerrilMendozaSantiago2020].

Among the techniques to induce cut cotorsion pairs, one has:

Proposition: Let \(\mathcal{A,B,S} \subseteq \mathcal{C}\) and \(\omega := \mathcal{A} \cap \mathcal{B}\) such that the following conditions are satisfied: (1) \(\mathcal{A}\) is closed under extensions and direct summands; (2) \(\mathcal{B}\) is closed under direct summands; (3) \(\omega \cap \mathcal{S}\) is a relative cogenerator in \(\mathcal{A}\) (meaning that every object in \(\mathcal{A}\) can be embedded into an object in \(\omega \cap \mathcal{S}\) with cokernel in \(\mathcal{A}\)); (4) \((\omega \cap \mathcal{S})^\wedge \subseteq \mathcal{B}\) (that is, every object with finite resolution dimension relative to \(\omega \cap \mathcal{S}\) belongs to \(\mathcal{B}\)); and (5) \(\operatorname{Ext}^1_{\mathcal{C}}(\mathcal{A} \cap \mathcal{S},\mathcal{B}) = 0\) and \(\operatorname{Ext}^1_{\mathcal{C}}(\mathcal{A},\mathcal{B} \cap \mathcal{S}) = 0\). Then, \((\mathcal{A,B})\) is a cotorsion pair cut along \(\mathcal{A}^\wedge \cap \mathcal{S}\).

Proposition: If \(\mathcal{X} \subseteq \mathcal{C}\) is closed under extensions and direct summands, and if \(\omega \subseteq \mathcal{C}\) is an \(\mathcal{X}\)-injective relative cogenerator in \(\mathcal{X}\), then \((\mathcal{X},\omega^\wedge)\) is a left cotorsion pair cut along \(\mathcal{X}^\wedge\). If in addition, \(\omega\) is closed under extensions and direct summands, then \((\mathcal{X},\omega^\wedge)\) is also a right cotorsion pair cut along \(\mathcal{X}^\wedge\).

These methods can be applied, for instance, to obtain the following cut: Let \(\Lambda\) be the quotient path \(k\)-algebra of the quiver !quiver.jpeg|175x75 with relations \(\alpha \beta = 0 = \beta \alpha\). Then, for the Frobenius subcategory \(\mathcal{X} := {\rm add}\left( \begin{array}{c} 1 \end{array} \oplus \begin{array}{c} 2 \\ 1 \end{array} \oplus \begin{array}{c} 2 \end{array} \oplus \begin{array}{c} 1 \\ 2 \end{array}\right)\) of \(\mathsf{mod}(\Lambda)\) and the proinjectives \(\mathcal{P}(\mathcal{X}) = {\rm add}\left(\begin{array}{c} 1 \\ 2 \end{array} \oplus \begin{array}{c} 2 \\ 1 \end{array} \right)\) in \(\mathcal{X}\), one has that \((\mathcal{X},\mathcal{P}(\mathcal{X})^\wedge)\) is a cotorsion pair cut along \(\mathcal{X}^\wedge\).

Among possible applications of cut cotorsion pairs in some subfields of algebra, including representation theory, we can mention the following:

(1) A characterization of the finitistic dimension conjecture: The following are equivalent for any two-sided Artinian ring \(R\) and \(n \geq 0\): (a) \({\rm findim}(R) \leq n\). (b) \((\mathcal{P}(R)^{\wedge}_n, \mathcal{G}^{<\infty})\) is a left cotorsion pair cut along \((\mathcal{P}(R)^{\wedge}_n \cup {}^{\perp_1}(\mathcal{G}^{<\infty})) \cap \mathsf{mod}(R)\), where \(\mathcal{G}^{< \infty} = (\mathcal{P}(R)^\wedge \cap \mathsf{mod}(R))^{\perp_1}\). \({\rm (c)}\( \)\mathcal{P}(R)_n^{\wedge} \cap \mathsf{mod}(R) = {}^{\perp_1}(\mathcal{G}^{<\infty}) \cap \mathsf{mod}(R)\). (d) \(\exists \mbox{ }\mathcal{S} \subseteq \mathsf{Mod}(R)\), with \(R \in \mathcal{S}\), such that \((\mathcal{P}(R), (\mathcal{G}^{<\infty})_n^\vee)\) is a right cotorsion pair cut along \(\mathcal{S}\). Here, \((\mathcal{G}^{<\infty})_n^\vee\) denotes the class of modules for which there is a coresolution of length \(n\) by objects in \(\mathcal{G}^{<\infty}\).

(2) Serre quotients admitting right adjoints: Let \(\mathcal{S} \subseteq \mathcal{C}\) be a Serre subcategory. If the Serre quotient functor \(Q \colon \mathcal{C} \longrightarrow \mathcal{C} / \mathcal{S}\) admits a right adjoint, then \((\mathcal{S},\mathcal{S}^{\perp_0} \cap \mathcal{S}^{\perp_1})\) is a right cotorsion pair cut along \(\mathcal{S}^{\perp_0} := \{ M \in \mathcal{C} \text{ : } \operatorname{Hom}_{\mathcal{C}}(S,M) = 0, \text{} \forall S \in \mathcal{S} \}\). If in addition, \(\mathcal{C}\) is cocomplete and \(\mathcal{S}\) is closed under coproducts, then the converse also holds.

(3) Cuts from extriangulated categories: Let \(\mathfrak{A}\) be a skeletally small extriangulated category [NakaokaPalu2019] with weak kernels, and denote by \(\mathsf{mod}(\mathfrak{A}^{\rm op})\) the category of finitely presented contravariant functors \(\mathfrak{A} \to \mathsf{Mod}(\mathbb{Z})\), \({\rm def}(\mathfrak{A}^{\rm op})\) the subcategory of Auslander's defects over \(\mathfrak{A}\) [Ogawa2019], and \({\rm lex}(\mathfrak{A}^{\rm op})\) the subcategory of left exact functors. If the quotient functor \(Q \colon \mathsf{mod}(\mathfrak{A}^{\rm op}) \longrightarrow \mathsf{mod}(\mathfrak{A}^{\rm op}) / {\rm def}(\mathfrak{A}^{\rm op})\) admits a right adjoint, then \(({\rm def}(\mathfrak{A}^{\rm op}),{\rm lex}(\mathfrak{A}^{\rm op}))\) is a right cotorsion pair cut along \(({\rm def}(\mathfrak{A}^{\rm op}))^{\perp_0}\). In the particular case where \(\mathfrak{A}\) is triangulated, and \((\mathcal{U,V})\) is a cotorsion pair in \(\mathfrak{U}\), then \(({\rm def}(\mathcal{U}^{\rm op}),{\rm lex}(\mathcal{U}^{\rm op}))\) is a cotorsion pair cut along \(({\rm def}(\mathcal{U}^{\rm op}))^{\perp_0}\) if, and only if, \((\mathcal{U,V})\) is a co-\(t\)-structure in \(\mathfrak{U}\)."

Baur, Karin (University of Leeds/University of Graz), Çanakçi, İlke (VU Amsterdam), Jacobsen, Karin M. (Hausdorff Research Institute For Mathematics) Kulkarni, Maitreyee (Max-Planck Institut für Mathematik) Todorov, Gordana (Northeastern University)

Let \(\tilde A_{p,q}\) be a non-oriented cyclic quiver with \(p\) clockwise arrows and \(q\) counterclockwise arrows. Let \(\Lambda\) be a cluster-tilted algebra of type \(\tilde A_{p,q}\). The AR quiver of \(\bmod \Lambda\) then contains two non-homogeneous tubes of rank \(p\) and \(q\) respectively.

We will use the specialized Caldero-Chapoton map on the two tubes. This sends all cluster variables of an associated cluster to 1; for the rigid modules, this can be thought of as sending the module to its number of submodules [CalderoChapoton2006]. The result is two interlacing grids of positive integers of the following form, where the grey numbers have been added for convenience.

!ICRA-Frieze|690x206

The horizontal sequences of integers are periodic, respectively of period at most \(p\) and \(q\) in the two tubes. For any ""diamond"" $\tiny{\begin{array}{rcl}

  & a &  \\
  b & &c \\
  & d &

\end{array}}\( we have \)ad-bc=1$.

Such interlacing grids are called *infinite periodic friezes* and are a generalisation of finite friezes as introduced in [ConwayCoxeter1973]. The way we obtained them raise some interesting questions:

* Can any infinite periodic frieze be obtained in this manner? * Are the two friezes we obtain here related? E.g. can we obtain one from the other?

The key to the answer to both of these questions lies in the realisation that infinite periodic friezes are related to triangulations of the annulus with marked points [BaurParsonsTschabold].

In short, the quiddity sequence of a frieze can be read off the triangulation of the annulus by counting the number of arcs incident to each marked point and adding 1. A quiver of cluster-tilted type \(\tilde A_{p,q}\) can similarly be read off by having the arcs of the triangulation correspond to the vertices and adding arrows between vertices corresponding to immediately neighbouring arcs.

Our paper contains a full description of the maps between the three classes of objects, here we restrain ourselves to including an example of a pair of friezes, the associated triangulation of a marked annulus and the associated quiver:

!ICRA-correspondence|690x276

The quiver we obtain on the right is the underlying quiver of \(\Lambda\) [AssemBruestle Charbonneau-JodoinPlamondon2010]. We get a one-to-one correspondence if we look at ""essential"" substructures of friezes and triangulations:

* A *skeletal frieze* is a frieze whose quiddity sequence does not contain one and is different from \((2, \ldots, 2)\). * A *skeletal triangulation* is a triangulation without any arcs going between marked points on the same boundary component.

Any frieze (triangulation) can be reduced to a skeletal frieze (triangulation) in an essentially unique manner. This reduction respects the relationship between friezes and triangulations.

Theorem [BaurCanackiJKulkarniTodorov] We have one-to-one correspondences between the following sets:
1. Skeletal infinite periodic friezes of period \(p\).
2. Pairs of skeletal infinite periodic friezes where the first frieze has period \(p\).
3. Skeletal triangulations of the annulus with \(p\) marked points on the outer border.
4. Non-oriented cyclic graphs with exactly \(p\) clockwise arrows.

By using this theorem (and its weaker, non-skeletal version) we can now answer the two questions posed earlier in the positive:

* Any infinite frieze of period \(p\) can be obtained by using the specialized Caldero-Chapoton map on some cluster-tilted algebra of type \(\tilde A_{p,q}\). * Given an infinite frieze of period \(p\) we can calculate, in an essentially unique manner, a frieze of some period \(q\) such that the pair of friezes come from using the specialized Caldero-Chapoton map on a cluster-tilted algebra of type \(\tilde A_{p,q}\)."

Charley Cummings (University of Bristol)

Let \(A\) be a finite dimensional algebra over a field. We consider the smallest triangulated subcategory of \(\mathcal{D}(A)\) that contains the injective modules and is closed under coproducts. If this subcategory is equal to \(\mathcal{D}(A)\) then we say that *injectives generate for \(A\)*. Recently, Rickard [Rickard2019] showed that if injectives generate for \(A\), then \(A\) satisfies the finitistic dimension conjecture. In [Cummings2020] we provide techniques to detect if injectives generate for rings obtained from various ring constructions.

There is extensive work into finding practical methods that can be used to identify algebras that satisfy the finitistic dimension conjecture. One such approach is to consider a collection of related algebras and ask whether, if some of the algebras satisfy the conjecture, do the others as well. This idea has been used for recollements of derived categories, [Happel1993], [ChenXi2017] and operations defined on the quiver of a quiver algebra [FullerSaorín1992], [GreenPsaroudakisSolberg2018], [BravoPaquette2020], to name a few. We apply the same philosophy to the property of 'injectives generate'.

Recall that a recollement of derived module categories \((R) = (\mathcal{D}({B}), \mathcal{D}({A}), \mathcal{D}({C}))\) is a collection of three rings with triples of adjoint functors, \((i^*,i_*,i^!)\) and \((j_!,j^*,j_*)\), satisfying the properties in [BeĭlinsonBernsteinDeligne1982]. We prove the following theorem relating recollements to injective generation.

Theorem

Let \((R) = (\mathcal{D}({B}), \mathcal{D}({A}), \mathcal{D}({C}))\) be a recollement of unbounded derived module categories.

\(\hspace{0.5em}\) (i) Suppose that \(i_* \colon \mathcal{D}(B) \rightarrow \mathcal{D}(A)\) preserves complexes that are quasi-isomorphic to a

\(\hspace{1.5em}\) bounded complex of injectives. \(\hspace{1.5em}\) (a) If injectives generate for both \(B\) and \(C\) then injectives generate for \(A\). \(\hspace{1.5em}\) (b) If injectives generate for \(A\) then injectives generate for \(C\).

\(\hspace{0.5em}\) (ii) Suppose that \(i_* \colon \mathcal{D}(B) \rightarrow \mathcal{D}(A)\) preserves compact objects.

\(\hspace{1.5em}\) (a) If injectives generate for both \(B\) and \(C\) then injectives generate for \(A\). \(\hspace{1.5em}\) (b) If injectives generate for \(A\) then injectives generate for \(B\).

Note that this theorem can be applied to the triangular matrix ring

$ \hspace{14.5em} A = \begin{pmatrix} C & {_CM_B}
0 & B \end{pmatrix}, $

since \(A\) induces a recollement \((R)\) where \(i_*\) preserves compact objects.

The above theorem focuses on recollements of ordinary algebras, however, techniques that are used to prove this result can also be applied to recollements of dg algebras. In particular, we use this setup to prove that injective generation interacts well with the 'vertex removal' operation considered in [GreenPsaroudakisSolberg2018].

Theorem

Let \(A\) be a finite dimensional algebra over a field and \(e \in A\) be an idempotent. Suppose that \({Ae}{\otimes_{eAe}^L}{eA}\) is bounded in cohomology, and that the semi-simple \(A\)-module \({(1-e)A}/{\text{rad}({(1-e)A}})\) has finite injective dimension. If injectives generate for \(eAe\) then injectives generate for \(A\).

There is a natural dual property to 'injectives generate'. Consider the smallest triangulated subcategory of the derived category that contains the projective modules and is closed under products. If the subcategory is the entire derived category then we say that *projectives cogenerate for the ring*. In [Cummings2020] we provide similar results for projective cogeneration to the ones stated here."

Jing Guo (USTC). Joint work with Yuming Liu (BNU), Yu Ye (USTC).

Brauer graph algebras coincide with symmetric special biserial algebras. Let \(A\) be a Brauer graph algebra associated with a Brauer graph \(G=(V(G),E(G),m),\) where \(V(G)\) is the vertex set, \(E(G)\) is the edge set and \(m\) is the multiplicity function of \(G.\) We denote by \(gr(A)\) the graded algebra associated with the radical filtration of \(A.\) Unlike \(A,\) there are less work on \(gr(A).\) In this work we will use the graded degrees of vertices in \(G\) to give a criterion for when \(gr(A)\) is representation-finite and when \(gr(A)\) is domestic.

Definition. Let \(G\) be a Brauer graph. For any \(v\in V(G),\) we define the graded degree \(grd(v)\) by \(\)

grd(v)=\begin{cases}
m(v)val(v), &    \mbox{if }m(v)val(v)>1,\\
m(v')val(v'), &   \mbox{if }m(v)val(v)=1,
\end{cases}

\(\) where \(val(v)\) is the valency (a loop is counted twice) of \(v,\) and \(v'\) is the unique vertex adjacent to \(v\) if \(val(v)=1.\)

We call an edge !i|153x37, 50% in \(G\) with \( grd(v_{1})\neq grd(v_{2})\) an unbalanced edge, and denote the endpoints of \(i\) by \(v_L^{(i)}\), \(v_S^{(i)}\) with \(grd(v_L^{(i)})>grd(v_S^{(i)}).\) Whenever the context is clear we will omit the superscript \((i).\) Moreover, if \(G\) is a Brauer tree and !i_SL|154x37, 50% is an unbalanced edge, we write the subgraph of \(G\) by removing the edge \(i\) as follows: \(G\setminus i=G_{i,L}\bigcup G_{i,S},\) where \(G_{i,L}\) (resp. \(G_{i,S}\)) is the connected branch of \(G\setminus i\) containing the vertex \(v_{L}\) (resp. \(v_{S}\)). We denote the set of vertices in \(G_{i,L}\) (resp. \(G_{i,S}\)) by \(V(G_{i,L})\) (resp. \(V(G_{i,S})).\)

Definition. Let \(G\) be a Brauer graph and \(u, v\in V(G),\( \)u\neq v.\) A walk from \(u\) to \(v\) is a sequence \([v_{1},a_{1},v_{2},\ldots, v_{k-1}, a_{k-1},v_{k}]\) of distinct vertices \(v_{1}=u,\ldots, v_{k}=v\) and distinct edges \(a_i\) incident to \(v_i\) and \(v_{i+1}\) for \(1\leq i\leq k-1.\) We say that a walk \([v_{1},a_{1},v_{2},\ldots, v_{k-1}, a_{k-1},v_{k}]\) is degree decreasing if \(grd(v_{1})\geq grd(v_{2})\geq\cdots\geq grd(v_{k}).\)

The length of a walk is defined to be the number of edges in the walk. When \(G\) is a tree, we denote by \(d_G(u,v)\) the length of the unique walk from \(u\) to \(v\) for distinct vertices \(u\) and \(v;\) a pair \((i,j)\) of unbalanced edges \(i, j\) in \(G\) is called an edge pair if \(j\) is an edge in \(G_{i,S}\) and \(d_G(v^{(j)}_S,v^{(i)}_S)+1=d_G(v^{(j)}_L,v^{(i)}_S).\) Let \(n_0\) be the number of unbalanced edges \(i\) in \(G\) such that \(v_0 \in V(G_{i,S})\) and \(n_1\) the number of edge pairs in \(G.\)

Theorem ([GuoLiu2020]). a). If \(gr(A)\) is of finite representation type, then \(G\) is a Brauer tree. b). Let \(G\) be a Brauer tree with an exceptional vertex \(v_0\) of multiplicity \(m_0.\) Then the following are equivalent. 1. \(gr(A)\) is of finite representation type. 2. Any walk from \(v\) is degree decreasing, where \(v=v_0\) when \(m_0>1\) or \(v\) is a vertex with maximal graded degree when \(m_0=1.\) 3. \(n_0(m_0-1)+n_1=0.\)

Remark. In case that \(gr(A)\) is representation-finite, there is a close connection between the Auslander-Reiten quiver of \(A\) and the Auslander-Reiten quiver of \(gr(A)\) ([GuoLiu2020]).

Theorem. a). \(gr(A)\) is 1-domestic if and only if one of the following holds 1. \(G\) is a Brauer tree with an exceptional vertex \(v_0\) of multiplicity \(m_0\) and \(n_0(m_0-1)+n_1=1.\) 2. \(G\) is a tree and there exist two distinct vertices \(i_0,i_1,\) such that 2.1. \(m(i_0)=m(i_1)=2\) and \(m(i)=1\) for \(i\neq i_0,i_1.\) 2.2. \(grd(i_0)=grd(i_1).\) 2.3. Any walk from \(i_0\) is degree decreasing. 3. \(G\) is a graph with a unique cycle of odd length and \(m(v)=1\) for all \(v\in V(G),\) and satisfies 3.1. \(grd(u)=grd(v)\) for any two vertices \(u\) and \(v\) in the unique cycle. 3.2. Any walk from any vertex in the unique cycle is degree decreasing.

b). \(gr(A)\) is 2-domestic if and only if \(G\) satisfies the following conditions. 1. \(G\) is a graph with a unique cycle of even length and \(m(v)=1\) for all \(v\in V(G).\) 2. \(grd(u)=grd(v)\) for any two vertices \(u\) and \(v\) in the unique cycle. 3. Any walk from any vertex in the unique cycle is degree decreasing.

c). \(gr(A)\) is not n-domestic for \(n\geq3.\)"

Zhen Zhang (BNU). Joint work with Aaron Chan (Nagoya U.), Jing Guo (USTC), Yuming Liu (BNU), and Yu Ye (USTC), based on [GLYZ1, 2020] and [CLZ, 2020].

Koenig and Liu [2011] introduced simple-minded system (s.m.s. for short) in the stable category of any artin algebra. Roughly speaking, an s.m.s. of an artin algebra \(A\) is a family of objects in the stable category \(A\!\(-\)\!\underline{\mathrm{mod}}\) which satisfies orthogonality and a generating condition. Chan, Koenig and Liu [2015] noticed that, for an indecomposable, basic representation-finite self-injective algebra (\(\ncong\!k\)) over an algebraically closed field \(k\) (RFS algebra for short), the s.m.s.'s in \(A\!\(-\)\!\underline{\mathrm{mod}}\) correspond exactly to the combinatorial configurations in the stable AR-quiver of \(A,\) a notion introduced by Riedtmann in the 1980's.

In general, it is hard to check the two conditions in the definition of an s.m.s., however we give an easy characterization of s.m.s.'s over RFS algebras.

Theorem 1. [GLYZ1, 2020]) Let \(A\) be an RFS algebra and \(\mathcal{S}\) a family of objects in \(A\!\(-\)\!\underline{\mathrm{mod}}.\) Then \(\mathcal{S}\) is an s.m.s. if and only if \(\mathcal{S}\) satisfies the following three conditions. (1) \(\mathcal{S}\) is a Hom-orthogonal system in \(A\!\(-\)\!\underline{\mathrm{mod}},\) that is, for any $S,T\in\mathcal{S},\( \)
{\rm \underline{Hom}}_A(S,T)\cong \left\{\begin{array}{ll} 0 & (S\neq T),
k & (S=T).\end{array}\right.$ (2) The cardinality of \(\mathcal{S}\) is equal to the number of non-isomorphic simple \(A\!\)-modules. (3) \(\mathcal{S}\) is Nakayama-stable, that is, the Nakayama functor \(D(A)\otimes_{A}-\) on \(A\!\(-\)\!\underline{\mathrm{mod}}\) permutes the objects of \(\mathcal{S}.\)

As an application, we present an explicit construction of s.m.s.'s over self-injective Nakayama algebras (cf. [GLYZ, 2020]). Note that our construction is independent with Riedtmann's (1980) and Chan's (2015) implicit ones.

We also study some behaviors of s.m.s.'s on a quasi-tube over a general self-injective algebra. The definition of a quasi-tube states as follows.

Definition 2. A component of AR-quiver of a self-injective algebra is called a quasi-tube of rank \(n\) (homogeneous tube, if \(n=1\!\)), if its stable part (by removing projective vertices) is of the form \(\mathbb{Z}A_\infty/\langle\tau^n\rangle,\) where integer \(n\geqslant 1.\)

Malicki and Skowronski [2011] showed that there are at most \(n-1\) simple modules on a quasi-tube of rank \(n\) over a self-injective algebra. The following result generalizes their conclusion.

Theorem 3. [CLZ, 2020] Let \(A\) be a self-injective algebra, \(\mathcal{C}\) a quasi-tube of rank \(n\) of \(A,\) and let \(\mathcal{S}\) be an s.m.s. Then (1) \(\mid \mathcal{S}\cap\mathcal{C}\!\mid< n.\) In particular, none of the indecomposable module in an s.m.s. of \(A\) lie in any homogeneous tube. (2) No object of \(\mathcal{C}\) with quasi-length \(\geq n\) belongs to \(\mathcal{S}.\)

It was shown by Crawley-Boevey that, if \(A\) is tame, then "almost all modules" lie in homogeneous tubes. Therefore our result excludes most of the modules of a tame self-injective algebra from forming an s.m.s."

Andrzej Mróz (Nicolaus Copernicus University) Bartosz Makuracki (Nicolaus Copernicus University)

Quasi-Cartan matrices were introduced by [BarotGeissZelevinsky2006] in the context of cluster algebras. They generalize the notion of a Cartan matrix which arises from the root system associated with a complex semisimple Lie algebra and which plays a fundamental role in Lie theory. We recall that a quasi-Cartan matrix is an integer matrix \(A=[[https://www.cambridge.org/core/books/infinitedimensional-lie-algebras/053FE77E6E9B35C56B5AEF7336FE7306|a_{ij}]\in\mathbb{M}_n(\mathbb{Z})\) such that \(a_{ii}=2\) for \(i=1,\ldots,n\) and \(A\) is symmetrizable, that is, \(DA\) is symmetric for some diagonal \(D\in\mathbb{M}_n(\mathbb{Z})\) with positive diagonal entries. Moreover, \(A\) is called a (generalized) Cartan matrix in case \(a_{ij}\leq 0\) for \(i\neq j\). These matrices appear in some variants in many contexts of Lie theory, cluster theory and in the study of quadratic forms and bilinear lattices associated to fin-dim associative algebras, see e.g.: Kac1990], [BarotKussinLenzing2006], [PerezRivera2019], [LenzingdelaPeña2008], [Mróz2016], [MrózdelaPeña2016], [Simson2013] and [Simson2020].

It is known that each (connected) positive definite quasi-Cartan matrix \(A\) is \(\mathbb{Z}\)-equivalent with the Cartan matrix of a Dynkin diagram, called the Dynkin type of \(A\). We note that the \(\mathbb{Z}\)-equivalence relation \(A\sim A'\) (\(\Leftrightarrow\ \ M^{\rm tr}DAM = D'A'\) for some \(M\in{\rm Gl}_n(\mathbb{Z})\) and \(D'=D\) up to permutation) corresponds to the isomorphism of associated semisimple Lie algebras, see [PerezRivera2019].

Efficient algorithmic methods to compute the Dynkin type of a positive definite quasi-Cartan matrix can be of importance for computational approach to the combinatorial aspects of the above mentioned areas. Some (implicit of explicit) algorithms for this task are known. Recall that ""roughly estimated"" classical [Ovsienko1978]'s methods based on so-called inflations (Gabrielov transformations) lead to exponential algorithms. Recently some more efficient methods were discovered, e.g. * for \(A\) symmetric: [AbarcaRivera2016] (algorithm of the arithmetic complexity \(\mathcal{O}(n^5{\rm log}^2n)\) ), [DräxlerDrozdGolovachtchukOvsienkoZeldych1995] (\(\mathcal{O}(n^4)\)), [Mróz2016] (\(\mathcal{O}(n^3)\)), * for an arbitrary quasi-Cartan \(A\) (satisfying some technical assumptions): [PerezAbarcaRivera2018] (\(\mathcal{O}(n^3)\) - the best known algorithm so far…).

Recently we obtained in [MakurackiMróz2021] the following improvement of the above methods.

Theorem 1. *There exists an algorithm to compute the Dynkin type of a positive definite (not necessarily symmetric) quasi-Cartan matrix, of the arithmetic computational complexity* \(\mathcal{O}(n^2)\).

We prove the theorem by providing an explicit new variant of the inflation algorithm. We note that the obtained complexity seems to be the lowest possible for this problem, since it is asymptotically the same as simply reading the input matrix. In the construction of our algorithm as well as in the estimation of its arithmetic complexity we apply, among others: * the recent (quite simple) observation from [MakurackiMróz2019] (see also [Simson2015]) stating that there is a bijection between quasi-Cartan matrices and the so-called Cox-regular bigraphs in the sense of [KasjanSimson2015]; this allows us to adapt some of the useful techniques of algebraic graph theory developed for bigraphs in [Simson2013], [Simson2018] and [Simson2019]; * the study of the behaviour of the special irreducible root systems (in the sense of [Bourbaki1968]) constructed in [MakurackiMróz2019] by means of certain Coxeter orbits of roots of quadratic forms associated with positive definite quasi-Cartan matrices (cf. also [Simson2019] and [Simson2020]).

The algorithm can be applied to an arbitrary quasi-Cartan matrix (i.e., not necessarily positive definite), thus we immediately obtain the following:

Corollary 1. *There exists an algorithm to test if an arbitrary quasi-Cartan matrix is positive definite, of the arithmetic computational complexity* \(\mathcal{O}(n^2)\).

In this way we obtain a positive definiteness test more efficient than standard tests based on (naive) Gaussian elimination and Sylvester's criterion. Recall that the standard algorithm has the arithmetic complexity \(\mathcal{O}(n^3)\) and the exponential bit-complexity (also in restriction to quasi-Cartan matrices). Note that since in our algorithm we operate on integer matrices with bounded coefficients it is easy to see that the bit-complexity of our algorithm is \(\mathcal{O}(n^2{\rm log}_2n)\).

As another application of Theorem 1 we apply in [MakurackiMróz2021] our algorithm to support certain conceptual and computational classification of (symmetric and non-symmetric) quasi-Cartan matrices related to Nakayama algebras."

Nicholas Williams (University of Leicester)

In [Wil20], we show that the relationship discovered in [OT12] between triangulations of even-dimensional cyclic polytopes and the representation theory of the higher Auslander algebras of type \(A\) is richer than previously known. We show that triangulations of odd-dimensional cyclic polytopes not only enter the picture as well, but two orders on the set of triangulations of a cyclic polytope, the higher Stasheff–Tamari orders, translate naturally to the algebraic context.

This gives new insight into the representation theory of the higher Auslander algebras of type \(A\) and a new avenue for attacking the conjectured equivalence of the higher Stasheff–Tamari orders, a 24-year open problem. It also gives a new approach for studying the set of maximal green sequences of an algebra: under a natural equivalence relation, this set becomes a poset.

The higher Stasheff–Tamari orders

The two higher Stasheff–Tamari orders are two orders on the set of triangulations of a cyclic polytope introduced in [KV91][ER96]. The \emph{cyclic} polytope \(C(m,\delta)\) is defined to be the convex hull of \(m\) points on the moment curve \(p(t)=(t,t^{2}, \dots, t^{\delta})\).

The *first higher Stasheff–Tamari order* (\(\leqslant_{1}\)) is the poset with increasing bistellar flips as covering relations. In two dimensions, an increasing bistellar flip consists of replacing the diagonal \(13\) with the diagonal \(24\) inside a square labelled \(1234\).

There is a natural projection \(C(m,\delta+1) \rightarrow C(m,\delta)\) given by forgetting the last coordinate of \(\mathbb{R}^{\delta + 1}\). Every triangulation induces a section of this projection map. Given two triangulations \(\mathcal{T}, \mathcal{T}'\) of \(C(m,\delta)\), we have that \(\mathcal{T}\) is less than \(\mathcal{T}'\) with respect to the *second higher Stasheff–Tamari order* (\(\leqslant_{2}\)) if and only if the section induced by \(\mathcal{T}'\) lies above the section induced by \(\mathcal{T}\).

Edelman and Reiner conjecture that the two higher Stasheff–Tamari orders are equal [ER96], but this is still an open problem.

Higher Auslander–Reiten theory

Higher Auslander–Reiten theory was introduced by Iyama as a higher-dimensional generalisation of classical Auslander–Reiten theory [Iya07]. It studies \(d\)-cluster-tilting subcategories of \(\mathrm{mod}\,\Lambda\), where \(\Lambda\) is a finite-dimensional algebra. A functorially finite subcategory \(\mathcal{M}\) of \(\mathrm{mod}\,\Lambda\) is called *\(d\)-cluster-tilting* if

$ \mathcal{M} = \{ X \in \mathrm{mod}\,\Lambda \mid \forall i \in [d-1], \forall M \in \mathcal{M}, \mathrm{Ext}_{\Lambda}^{i}(X,M) = 0 \} $ $ \quad \, \, \, = \{ X \in \mathrm{mod}\,\Lambda \mid \forall i \in [d-1], \forall M \in \mathcal{M}, \mathrm{Ext}_{\Lambda}^{i}(M,X) = 0 \}. $

If \(\mathrm{add}\,M\) is a \(d\)-cluster-tilting subcategory, for \(M \in \mathrm{mod}\,\Lambda\), then we say that \(M\) is a *\(d\)-cluster-tilting module*.

The canonical examples of algebras with \(d\)-cluster-tilting modules are the higher Auslander algebras of type \(A\), denoted \(A_{n}^{d}\), which were introduced by Iyama in [Iya11]. Here the algebra \(A_{n}^{1}\) is the path algebra of linearly oriented \(A_{n}\), and \(A_{n}^{d+1} \cong \mathrm{End}_{A_{n}^{d}} M^{(d,n)}\), where \(M^{(d,n)}\) is the unique basic \(d\)-cluster-tilting module in \(\mathrm{mod}\, A_{n}^{d}\).

Results

Oppermann and Thomas show that triangulations of \(C(n+2d,2d)\) are in bijection with basic tilting \(A_{n}^{d}\)-modules with summands in \(\mathrm{add}\, M^{(d,n)}\). We show that the higher Stasheff–Tamari orders correspond to the orders on tilting modules studied in [RS91][HU05].

Theorem 1 (Theorem D, [Wil20]) *Let \(\mathcal{T}\) and \(\mathcal{T}'\) be triangulations of \(C(n+2d,2d)\) corresponding to tilting \(A_{n}^{d}\)-modules \(T\) and \(T'\). We then have that*

1. *\(\mathcal{T} \lessdot_{1} \mathcal{T}'\) if and only if \(T'\) is a left mutation of \(T\); and*

2. *\(\mathcal{T} \leqslant_{2} \mathcal{T}'\) if and only if \(^{\bot}T \subseteq {}^{\bot}T'\).*

There exists an analogue of this theorem which considers cluster-tilting objects instead of tilting modules. We further prove that triangulations of \(C(n+2d+1,2d+1)\) are in bijection with equivalence classes of \(d\)-maximal green sequences of \(A_{n}^{d}\). We then show how the higher Stasheff–Tamari orders may be interpreted here.

Theorem 2 (Theorem E, [Wil20]) *Let \(\mathcal{T}\) and \(\mathcal{T}'\) be triangulations of \(C(n+2d+1,2d+1)\) corresponding to equivalence classes of \(d\)-maximal green sequences \([G]\) and \([G']\) of \(A_{n}^{d}\). We then have that*

1. *\(\mathcal{T} \lessdot_{1} \mathcal{T}'\) if and only if \([G']\) is an increasing elementary polygonal deformation of \([G]\); and*

2. *\(\mathcal{T} \leqslant_{2} \mathcal{T}'\) if and only if the set of summands of \([G]\) contains the set of summands of \([G']\).*

The equivalence relation holds when \(d\)-maximal green sequences have the same set of indecomposable summands. Modulo this equivalence relation, the set of maximal green sequences of \(A_{n}\) is a lattice, by [ER96]."

Fang Li (Zhejiang University)

Joint work with *Zhihao Wang, Jie Wu, Bin Yu.*

Let \(S\) be a semigroup and \(M\) a non-empty topological space. If the map

\(L_S: M\longrightarrow M\) satisfies \(L_{s_{2}s_{1}}(m)=L_{s_{2}}\big(L_{s_{2}}(m)\big)\)

%\(\varphi(s_{2},\varphi(s_{1},m))=\varphi(s_{2}s_{1},m)\),

$\forall s_{1},s_{2}\in S\(, \)\forall m\in M\(, then \)(M, L_S)$ is called

a left \(S\)-topological system. Let \(M,N\) be two \(S\)-Systems of topological spaces, a continuous map \(f:M\longrightarrow N\) is called an

\(S\)-morphism from \(M\) to \(N\), if \(f(sm)=sf(m)\), \(\forall s\in S\) and \(\forall m\in M\). All left \(S\)-topological systems and all

\(S\)-morphisms between them constitute a category, denoted by

\(S\)-\(\mathcal{TOP}\).

Let \(\Gamma=(\Gamma_{0},\Gamma_{1})\) be a quiver with \(\Gamma_{0}\)

the set of vertices and \(\Gamma_{1}\) the set of arrows between

vertices. A topological representation \((T,f)\) of a quiver

\(\Gamma=(\Gamma_{0},\Gamma_{1})\) is a family of pairs of topological space $\{T_{i}: i\in\Gamma_{0}\}\( together with continuous map \)f_{\alpha}$:

\(T_{i}\rightarrow T_{j}\) for each arrow \(\alpha\): \(i\rightarrow j\). Let \((T, f)\) and \((T', f')\) be two topological representations of \(\Gamma\).

A morphism \(h\): \((T,f)\rightarrow(T',f')\) is a collection of continuous maps

\(\{h_{i}:T_{i}\rightarrow T'_{i}\}_{i\in\Gamma_{0}}\) such

that for each arrow \(\alpha\): \(i\rightarrow j\) in \(\Gamma_{1}\) the standard commutative diagram.

Denote by \(\textbf{Top}\mathrm{-}\textbf{Rep}\Gamma\) the category consisting of topological representations of a quiver \(\Gamma\) and the morphisms between the topological representations.

Theorem. The categories \(\textbf{Top}\mathrm{-}\textbf{Rep}\Gamma\) and \(P(\Gamma)\(-\)\mathcal{TOP}^{o}\) are equivalent, where \(P(\Gamma)\) is the semigroup related to the quiver \(\Gamma\) and \(P(\Gamma)\(-\)\mathcal{TOP}^{o}\) is a full subcategory of \(P(\Gamma)\(-\)\mathcal{TOP}\).

Based on the known works, we investigate the relation between the category of topological representations and that of linear representations of a quiver via \(P(\Gamma)\(-\)\mathcal{TOP}^o\) and \(k\Gamma\)-Mod.

Theorem. Let \(\Gamma\) be a finite connected quiver and \(k\) a field, then the following statements are equivalent:

(i) all \(k\Gamma\)-modules are (positively) graded;

(ii) all \(P(\Gamma)\)-Systems in \(P(\Gamma)\(-\)\mathcal{TOP}^{o}\)

are (positively) graded.

The similar result holds for their vertex (positively) graded versions. We also defined the homology groups of topological representations and homotopy equivalence on topological representations. On these definitions, the Homotopy Axiom holds. Moreover, many topological properties of a quiver can be read from these homology groups.

Theorem. Let \((T,f),(T^{'},f^{'})\in\textbf{Top}\mathrm{-}\textbf{Rep}\Gamma\), and \(\mu\simeq^{t}\nu:(T,f)\rightarrow(T^{'},f^{'})\). Then \(S^{\Gamma}(\mu)\simeq^{t} S^{\Gamma}(\nu):S^{\Gamma}(T,f)\rightarrow S^{\Gamma}(T^{'},f^{'})\), and therefore \(H^{\Gamma}_n(\mu)=H^{\Gamma}_n(\nu)\) for all \(n\geq0\).

Theorem. Let \(\Gamma\) be a connected quiver, and \((T,f)\) be a top-representation with \(T(i)=S^{1},f_{ij}=F\). Then \(H_1(T,f)=\mathbb{Z}\) if \(\Gamma\) is acyclic, and \(H_1(T,f)=0\) if \(\Gamma\) contains oriented circle.

Theorem. \(H_0(\frac{S(S^1)}{ImL})\simeq\mathbb{Z}_2\) if \(\Gamma\) contains at least one oriented circle whose length is odd; otherwise, \(H_0(\frac{S(S^1)}{ImL})=0\). We defined the functor \(At^{\Gamma}\) from the category \(\textbf{Top}\mathrm{-}\textbf{Rep}\Gamma\) to the category \(\textbf{Top}\). We have shown that \(At^{\Gamma}\) preserves homotopy equivalence between morphisms. Moreover, we discussed some properties of the functor \(At^{\Gamma}\) and established the connection between the homotopy groups of a top-representation \((T,f)\) and the homotopy groups of \(At^{\Gamma}(T,f)\).

Theorem. Let \(\Gamma\) be a quiver. For any \(\alpha\simeq^t\beta:(T,f)\rightarrow(T^{'},f^{'})\) in \(\textbf{Top}\mathrm{-}\textbf{Rep}\Gamma\). Then, as both morphisms from \(At^{\Gamma}(T,f)\) to \(At^{\Gamma}(T^{'},f^{'})\) in \(\textbf{Top}\), it holds \(At^{\Gamma}(\alpha)\simeq^t At^{\Gamma}(\beta)\).

Theorem. Let \(\Gamma\) be a connected quiver, and \((T,f)\in\textbf{Top}\mathrm{-}\textbf{Rep}\Gamma\) with all \(T(i)\) being connected. Then, \(At^{\Gamma}(T,f)\) is connected.

Theorem. For any quiver \(\Gamma\) with finite components, each \(n\geq0\), we have a natural tranformation from \(H_n(-)\) to \(H_nAt^{\Gamma}(-)\)."

Johanne Haugland (Norwegian University of Science and Technology)

*This research snapshot is based on [Haugland2019].*

Higher homological algebra is the theory of \(n\)-abelian, \(n\)-exact and \((n+2)\)-angulated categories. The investigation of such structures was initiated by Iyama [Iyama2007, Iyama2011], and axiomatizations were given in [Jasso2016] and [GeissKellerOppermann2013]. As a recent contribution to this active field of research, [HerschendLiuNakaoka2017] introduced the more general notion of \(n\)-exangulated categories, giving a higher dimensional analogue of extriangulated categories [NakaokaPalu2019]. Through the study of intrinsic properties of \(n\)-exangulated categories, we obtain unified knowledge about \(n\)-exact, \(n\)-abelian and \((n+2)\)-angulated categories, as well as proper generalizations of such.

Just as short exact sequences and distinguished triangles play a fundamental role in the study of abelian, exact and triangulated categories, an \(n\)-exangulated category comes equipped with an important class of distinguished \((n+2)\)-term sequences. These sequences are called conflations.

Let \(\mathcal{C}\) be an essentially small \(n\)-exangulated category and consider the free abelian group \(\mathcal{F}(\mathcal{C})\) generated by isomorphism classes \(\langle A \rangle\) of objects \(A\) in \(\mathcal{C}\). Given a conflation \[X_\bullet \colon~ X_0 \rightarrow X_1 \rightarrow \cdots \rightarrow X_n \rightarrow X_{n+1}\] in \(\mathcal{C}\), the corresponding Euler relation is the alternating sum of isomorphism classes \[\chi(X_\bullet)= \langle X_0 \rangle - \langle X_1 \rangle + \cdots + (-1)^{n+1}\langle X_{n+1} \rangle.\] This allows us to define the Grothendieck group of \(\mathcal{C}\).

Definition: The *Grothendieck group* of \(\mathcal{C}\) is the quotient \(K_0(\mathcal{C})=\mathcal{F}(\mathcal{C})/\mathcal{R}(\mathcal{C}),\) where \(\mathcal{R}(\mathcal{C})\) is the subgroup generated by the subset \(\{ \chi(X_\bullet) \mid X_\bullet\) is a conflation in \(\mathcal{C}\) \} if \(n\) is odd and \(\{\langle 0 \rangle \}\cup\{ \chi(X_\bullet) \mid X_\bullet\) is a conflation in \(\mathcal{C}\) \} if \(n\) is even.

In the study of an algebraic structure, a natural strategy is to investigate its substructures, thereby extending the understanding of the structure we started with. The following result gives a classification of certain subcategories of an \(n\)-exangulated category in terms of subgroups of the associated Grothendieck group. The subgroup \(H_{\mathcal{G}} \leq K_0(\mathcal{C})\) is generated by the set of isomorphism classes of objects in \(\mathcal{G}\subseteq\mathcal{C}\).

Theorem: Let \(\mathcal{C}\) be an \(n\)-exangulated category with \(n\) odd. Let \(\mathcal{G}\) be an \(n\)-(co)generator of \(\mathcal{C}\). There is then a one-to-one correspondence between subgroups of \(K_0(\mathcal{C})\) containing \(H_{\mathcal{G}}\) and dense complete subcategories of \(\mathcal{C}\) containing \(\mathcal{G}\).

Completeness is a higher dimensional analogue of what we know as the \(2/3\)-property for triangulated subcategories. The theorem above unifies and extends results from [Thomason1997, BerghThaule2014, Matsui2018, ZhuZhuang2019].

We show that the subcategories in our classification theorem inherit an \(n\)-exangulated structure from the ambient category. This makes them examples of *\(n\)-exangulated subcategories* (or *extriangulated subcategories* if \(n=1\)), as defined in [Haugland2019, Definition 3.7]. In this definition, we use the notion of an \(n\)-exangulated functor, as introduced by [BennettTennenhausShah]."

Laertis Vaso (Max Planck Institute for Mathematics) *This research snapshot is based on the preprint [Vaso2020b]*

Let \(\mathbf{k}\) be a field and \(\Lambda\) be a finite-dimensional \(\mathbf{k}\)-algebra. One of the main aims of representation theory is to describe the category \(\operatorname{mod}\Lambda\) of finite-dimensional (right) \(\Lambda\)-modules. Since it is generally impossible to achieve this aim, one may restrict to suitable subcategories of the module category. In higher dimensional Auslander–Reiten theory ([Iyama2008]) one changes focus from \(\operatorname{mod}\Lambda\) to a functorially finite subcategory \(\mathcal{M}\subseteq \operatorname{mod}\Lambda\) such that \(\) \begin{align*} \mathcal{M} &= \{ X \in \operatorname{mod}\Lambda \mid \operatorname{Ext}^{i}_{\Lambda}\left(X, \mathcal{M}\right)=0\text{ for all \(1\leq i \leq n-1\)}\}

  & = \{ X \in \operatorname{mod}\Lambda \mid \operatorname{Ext}^{i}_{\Lambda}\left(\mathcal{M},X\right)=0\text{ for all \(1\leq i \leq n-1\)}\}, 

\end{align*} \(\) called an \(n\)*-cluster tilting subcategory*. In this work we present a new method of constructing \(n\)-cluster tilting subcategories based, among other things on [Vaso2020a] and [DarpöIyama2020]).

Let \(\Lambda\) be a representation-directed algebra, that is a representation-finite algebra with no oriented cycles in its Auslander–Reiten quiver. A characterization of \(n\)-cluster tilting subcategories for representation-directed algebras is given in [Vaso2019]. Based on this characterization, a generalization of \(n\)-cluster tilting subcategories for representation-directed algebras called \(n\)*-fractured subcategories* is given in [Vaso2020a]. In [Vaso2020a] a gluing procedure is also introduced which takes as input two representation-directed algebras \(A\) and \(B\), together with a certain \(A\)-module and a certain \(B\)-module, and produces a new representation-directed algebra \(B{\scriptstyle\triangle} A\). The representation theory of \(B{\scriptstyle\triangle} A\) can be completely described via the representation theory of \(A\) and \(B\). If moreover \(\operatorname{mod}A\) and \(\operatorname{mod}B\) each admit an \(n\)-fractured subcategory \(\mathcal{M}_A\) and \(\mathcal{M}_B\), and some reasonable compatibility conditions between \(\mathcal{M}_A\) and \(\mathcal{M}_B\) are satisfied, then \(\operatorname{mod}(B{\scriptstyle\triangle} A)\) also admits an \(n\)-fractured subcategory \(\mathcal{M}_{B{\scriptscriptstyle \triangle} A}\). In many cases \(\mathcal{M}_{B{\scriptscriptstyle \triangle} A}\) is an actual \(n\)-cluster tilting subcategory.

As a first step, we generalize the above situation. Let \(G\) be a directed tree such that every vertex in \(G\) is the source and target of finitely many arrows. We denote the set of vertices of \(G\) by \(V_G\) and the set of arrows of \(G\) by \(E_G\). We decorate each vertex \(v\in V_G\) by a representation-directed bound quiver algebra \(\Lambda_v\) in such a way that each arrow \(e:u\to v\) of \(G\) corresponds to a gluing \(\Lambda_u{\scriptstyle \triangle}\Lambda_v\). We also introduce assumptions which have the effect that for every finite connected subgraph \(H\) of \(G\), we may perform the gluings induced by the arrows of \(H\) in any order to obtain a representation-directed algebra \(\Lambda_H\). This is the data of a *gluing system* \((\Lambda_v)_{v\in V_G}\). Both the algebra \(\Lambda_H\) and the category \(\operatorname{mod}\Lambda_{H}\) can be described completely via the algebras \(\Lambda_v\) for \(v\in V_H\).

Such a gluing system gives rise to a (potentially infinite) quiver \(\dot{Q}\) and a two-sided ideal \(\dot{\mathcal{R}}\) of its path category \(\mathbf{k} \dot{Q}\). Then the category \(\operatorname{mod}(\operatorname{proj} (\mathbf{k}\dot{Q}/\dot{\mathcal{R}}))\) can be described via the categories \(\operatorname{mod}\Lambda_v\) for \(v\in V_G\). In this way we may consider \(\operatorname{proj}(\mathbf{k} \dot{Q}/\dot{\mathcal{R}})\) as a suitable candidate for the limit of the system \((\Lambda_v)_{v\in V_G}\).

Assume now that the module category \(\operatorname{mod}\Lambda_v\) of each algebra \(\Lambda_v\) in our gluing system admits an \(n\)-fractured subcategory \(\mathcal{M}_v\subseteq \operatorname{mod}\Lambda_v\), and that these \(n\)-fractured subcategories are compatible with the gluing. This is the data of an \(n\)*-fractured system*. This data gives rise to a subcategory \(\dot{\mathcal{M}}\subseteq \operatorname{mod} (\mathbf{k}\dot{Q}/\dot{\mathcal{R}})\). In general \(\dot{\mathcal{M}}\) is not an \(n\)-cluster tilting subcategory. To remedy this we introduce the notion of *complete* \(n\)-fractured systems. We then have the following theorem.

Theorem 1 Let \(\mathcal{L}=\left(\Lambda_v\right)_{v\in V_G}\) be a gluing system and \(\left(\mathcal{M}_v\right)_{v\in V_G}\) be a complete \(n\)-fractured system of \(\mathcal{L}\). Then \(\dot{\mathcal{M}}\subseteq \operatorname{mod}(\mathbf{k}\dot{Q}/\dot{\mathcal{R}})\) is an \(n\)-cluster tilting subcategory.

In fact our result is quite more general. Instead of considering algebras and modules over algebras we consider the more abstract setting of inverse systems of Krull–Schmidt categories and modules over categories and prove our results in that setting. Theorem 1 becomes then a straightforward application of that setting.

Although starting from a complete \(n\)-fractured system we obtain an \(n\)-cluster tilting subcategory of \(\operatorname{mod}(\mathbf{k} \dot{Q}/\dot{\mathcal{R}})\), this is not, in general, an \(n\)-cluster tilting subcategory of the module category of a finite-dimensional algebra. To remedy this, we use the results of [DarpöIyama2020] and apply an orbit construction to \(\operatorname{mod}(\mathbf{k}\dot{Q}/\dot{\mathcal{R}})\). First assume that \(G\) is the graph \[\cdots \xrightarrow{\alpha_{-3}} -2 \xrightarrow{\alpha_{-2}} -1 \xrightarrow{\alpha_{-1}} 0 \xrightarrow{\alpha_{0}} 1 \xrightarrow{\alpha_{1}} 2 \xrightarrow{\alpha_{2}} \cdots,\] and that \(\Lambda_v\) is the same algebra for every \(i\in V_G\) and each arrow \(\alpha_i\in E_G\) corresponds to the same gluing. Then we define an admissible \(\mathbb{Z}\)-action on \(\mathbf{k}\dot{Q}/\dot{\mathcal{R}}\) which induces a \(\mathbb{Z}\)-action on \(\operatorname{proj}(\mathbf{K} \dot{Q}/\dot{\mathcal{R}})\). Next assume that \(\mathcal{M}_v\subseteq \operatorname{mod}\Lambda_v\) is the same \(n\)-fractured subcategory for every \(v\in V_G\) and that the pair \((\Lambda_v,\mathcal{M}_v)\) satisfies certain symmetry conditions; in this case we call \(\Lambda_v\) an \(n\)*-self-gluable* algebra. Then this data gives rise to an \(n\)-cluster tilting subcategory \(\tilde{M}\) of \(\operatorname{mod}\big(\operatorname{proj}(\mathbf{k} \dot{Q}/\dot{\mathcal{R}})/\mathbb{Z}\big)\) by Theorem 2.13 in [DarpöIyama2020]). We then show the following.

Theorem 2 There is an equivalence of categories \(\operatorname{mod}\big(\operatorname{proj}(\mathbf{k} \dot{Q}/\dot{\mathcal{R}})/\mathbb{Z}\big)\simeq \operatorname{mod} \tilde{\Lambda}\) where \(\tilde{\Lambda}\) is a finite-dimensional bound quiver algebra. In particular, \(\operatorname{mod}\tilde{\Lambda}\) admits an \(n\)-cluster tilting subcategory \(\tilde{M}\).

In this way we obtain an \(n\)-cluster tilting subcategory for a finite-dimensional algebra. Moreover, this new algebra is not, in general, representation-directed, although it is still representation-finite.

Using the above methods we construct many new families of algebras whose module categories admit an \(n\)-cluster tilting subcategory and which have many interesting properties. As an example we prove the following theorem.

Theorem 3 For all \(s,t\in\mathbb{Z}_{\geq 0}\) there exists a bound quiver algebra \(\Lambda=\mathbf{k}Q/\mathcal{R}\) such that \(Q\) has \(s\) sources and \(t\) sinks and such that \(\operatorname{mod}\Lambda\) admits an \(n\)-cluster tilting subcategory."

Raphael Bennett-Tennenhaus (University of Leeds / Bielefeld University) *based on* [Bennett-Tennenhaus2019] *and* [Bennett-Tennenhaus2020].

A fundamental idea in model theoretic algebra is to study algebraic objects, such as modules, by looking at the formulas they satisfy. Baur's result on quantifier elimination reduced the focus to pp formulas, and motivated the study of pure-injective and \(\Sigma\)-pure-injective modules. Among the ensuing results were various characterisations of such modules in both categorical and model-theoretic terms, which proved to be invaluable as the subject evolved.

Krause [Krause2002] adapted the notions of pure-monomorphisms and pure-injective objects to the setting of compactly generated triangulated categories, and gave characterisations of pure-injective objects which mimic those discussed above. Garkusha and Prest [GarkushaPrest2005] then introduced a multi-sorted language for these categories, which behaves analogously to the language of modules over a fixed ring. The first of our two main results is the following characterisation of the \(\Sigma\)-pure-injective objects in this setting.

<blockquote>

<strong>Theorem.</strong> Let \(\mathcal{T}\) be a compactly generated triangulated category with a set \(\mathtt{G}\) of generators and whose subcategory \(\mathcal{T}^{c}\) of compact objects is skeletally-small. Let \(\mathbf{Y}:\mathcal{T}\to\mathbf{Mod}\text{-}\mathcal{T}^{c}\) be the restricted Yoneda functor. For an object \(M\) of \(\mathcal{T}\) the following statements are equivalent.

1. \(M\) is \(\Sigma\)-pure injective, that is, for any set \(I\) the coproduct \(M^{(I)}\) is pure-injective. 2. The countable coproduct \(M^{(\mathbb{N})}\) is pure-injective. 3. For any generator \(\mathscr{G}\in\mathtt{G}\) each ascending chain \(\bigcap_{\theta\in\mathtt{K}[1]}\mathrm{ker}(\theta)\subseteq \bigcap_{\theta\in\mathtt{K}[2]}\mathrm{ker}(\theta)\subseteq \dots\) of \(\mathbf{Y}(M)\)-annihilator subobjects of \(\mathscr{G}\) must stabilise. 4. For any set \(I\) the canonical morphism from \(M^{(I)}\) to the product \(M^{I}\) is a section. 5. For any object \(X\) of \(\mathcal{T}^{c}\) each descending chain \(\varphi_{1}(M)\supseteq \varphi_{2}(M)\supseteq\dots\) of pp-definable subgroups of \(M\) of sort \(X\) must eventually stabilise. 6. \(M\) is pure injective, and for any set \(I\) the object \(M^{I}\) is isomorphic to a coproduct of indecomposable pure-injective objects with local endomorphism rings.

</blockquote>

This result is analogous to the characterisation of \(\Sigma\)-pure-injective modules discussed above. The proof involves two approaches. In the first approach the one-sorted language of modules over a fixed ring is replaced with the canonical multi-sorted language introduced in [GarkushaPrest2005]. The second approach is to translate statements between the the categories \(\mathcal{T}\) and \(\mathbf{Mod}\text{-}\mathcal{T}^{c}\) using the restricted Yoneda functor, and take advantage of results on \(\Sigma\)-injective objects in locally coherent Grothendieck categories.

The characterisations discussed above have been used by a number of authors to classify pure-injectives in both module categories and triangulated categories. For an example of the former, Prest and Puninski [PrestPuninski2016] classified indecomposable pure-injective modules for domestic string algebras, verifying a conjecture of Ringel Ringel1995. For an example of the latter, Arnesen, Laking, Pauksztello and Prest [ArnesenLakingPauksztelloPrest2017] classified pure-injective indecomposable objects in the homotopy category of complexes of projective modules over derived-discrete algebras.

The goal of this project was to extend the classification from [ArnesenLakingPauksztelloPrest2017] to all gentle algebras by restricting our focus to \(\Sigma\)-pure-injective objects. The second of our two main results gives a description of \(\Sigma\)-pure-injective objects in the homotopy category of complexes of projectives over any gentle algebra.

<blockquote>

<strong>Theorem.</strong> Let \(k\) be a field, \(k[T,T^{-1}]\) the ring of Laurent polynomials, and \(\Lambda\) be a gentle algebra over \(k\). Every \(\Sigma\)-pure-injective object in the homotopy category \(\mathcal{K}(\Lambda\text{-}\bf{Proj})\) of complexes of projective modules is isomorphic to a coproduct of (shifts of) string complexes and band complexes paramenterised by \(\Sigma\)-pure-injective \(k[T,T^{-1}]\)-modules.

</blockquote>

This result is analogous to the description of \(\Sigma\)-pure-injective modules over (possibly non-domestic or infinite-dimensional) string algebras achieved in joint work with Crawley-Boevey [Bennett-TennenhausCrawley-Boevey2018]. Furthermore, the method of proof was also similar: both employing variations on the so-called functorial filtrations method from representation theory. Key differences and complications arose in the consideration of pp-definable subgroups of a complex (of a particular sort)."

Claire Amiot (Institut Fourier)

*Joint work with Thomas Brüstle*

Skew-gentle algebras have been introduced by Geiss and de la Peña in 1999 as skew-group algebras of gentle algebras equipped with a certain \({\mathbb Z}_2\)-action. We are using this skew-group algebra structure to describe the derived category of skew-gentle algebras.

OpperPlamondonSchroll2018 developed a geometric model for the derived category of gentle algebras, using a marked surface \((S,M)\) dissected into polygons. Independently, Haiden, Katzarkov and Kontsevich established a description of the (partially wrapped) Fukaya category of a surface \(S\) with stops \(M\) using the derived category of a (graded) gentle algebra associated to these data, given by a dissection of \(S\), and LekiliPolishchuk2018 discussed the derived equivalences for gentle algebras in this context. Combining results of OpperPlamondonSchroll2018 and LekiliPolishchuk2018, a geometric interpretation of the derived equivalence relation for gentle algebras is given in AmiotPlamondonSchroll2019 and Opper2019.

We extend these results to orbifolds \(\bar{S}\) admitting a two-fold cover. The two-fold cover \(S\) corresponds to a gentle algebra which comes equipped with a \({\mathbb Z}_2\)-action. We give a bijection between skew-gentle algebras and certain dissections that we call x-dissections of orbifolds admitting a double cover, showing that the \({\mathbb Z}_2\)-action on the double cover of the surface is compatible with the skew-group algebra construction. We further associate to each skew-gentle algebra a line field on the orbifold, and on its double cover, and interpret different kinds of derived equivalences of skew-gentle algebras in terms of diffeomorphisms respecting the homotopy class of the line fields associated to the algebras.

More precisely, to any dissected surface which is invariant under the action of an order-\(2\) diffeomorphism (with finitely many fixed points), we associate

* a gentle algebra \(\Lambda\) together with a \({\mathbb Z}_2\) action

* an orbifold surface together with a x-dissection

* a line field \(\bar{\eta}\) on the orbifold surface.

We then show that the skew-gentle algebra corresponding to the x-dissection is Morita equivalent to the skew-group algebra \(\Lambda {\mathbb Z}_2\), and that any skew-gentle algebra arises in this way.

Furthermore, we interpret geometrically two different notions of derived equivalence between skew-gentle algebras.

The first one is given by the existence of a \({\mathbb Z}_2\)-invariant tilting object.

Theorem: *Two skew-gentle algebras \(\bar{\Lambda}\) and \(\bar{\Lambda} '\) are derived equivalent via a \({\mathbb Z}_2\)-invariant tilting object if and only if there exists a diffeomorphism between their corresponding orbifolds sending \(\bar{\eta}\) to \(\bar{\eta}'\) up to homotopy.*

The second one is given by the existence of a \({\mathbb Z}_2\)-invariant tilting object with compatible action of \({\mathbb Z}_2\).

Theorem: *Two skew-gentle algebras \(\bar{\Lambda}\) and \(\bar{\Lambda}'\) are \({\mathbb Z}_2\)-derived equivalent, if and only if there exists a diffeomorphism between their corresponding 2-folded covers commuting with the \({\mathbb Z}_2\)-action and sending \(\eta\) to \(\eta'\) up to homotopy.*"

Qiu, Yu (Tsinghua University)

We introduce the \(q\)-deformation of topological Fukaya categories, Bridgeland stability conditions and quadratic differentials (of exponential type) in [IkedaQiuZhou2020], [IkedaQiu2018a] and [IkedaQiu2018b] (updated recently) respectively. For categories, we introduce the Calabi-Yau-\(\mathbb{X}\) category \(\mathcal{D}_{\mathbb{X}}(\mathbf{S}_\Delta)\) associated to a graded decorated marked surface \(\mathbf{S}_\Delta\), as the \(q\)-deformation of the topological Fukaya category \(\mathcal{D}_{\infty}(\mathbf{S}^\lambda)\) of a graded marked surface \(\mathbf{S}^\lambda\). We prove that the string objects and the morphisms in \(\mathcal{D}_{\mathbb{X}}(\mathbf{S}_\Delta)\) can be realized by curves and the intersections on \(\mathbf{S}_\Delta\). This result generlizes and unifies the corresponding results in the Calabi-Yau-3 setting in [Qiu16]/[QiuZhou2019] and in the Calabi-Yau-\(\infty\) setting in [HaidenKatzarkovKontsevich2017] (cf. [OpperPlamondonSchroll2018]). Note that here the Calabi-Yau-\(\infty\) category menas the derived category of some graded gentle algebra. Moreover, we introduce \(q\)-stability conditions on Calabi-Yau-\(\mathbb{X}\) categories, which consists of a Bridgeland stability condtion and a complex parameter \(s\) satisfying some Gepner equation. We show that the usual stability conditions on Calabi-Yau-\(\infty\) categories can induce \(q\)-stability conditions when the real part of \(s\) is bigger enough. By taking orbit categories, the \(q\)-stability conditions becomes usual stability conditions on Calabi-Yau-\(N\) categories. Finally, we introduce CY-\(s\) type \(q\)-quadratic differentials on \(\mathrm{log}\mathbf{S}_\Delta\), as \(q\)-deformation of quadratic differentials on \(\mathbf{S}^\lambda\). We show that the induced \(q\)-stability conditions on \(\mathcal{D}_{\mathbb{X}}(\mathbf{S}_\Delta)\), with fixed parameter \(s\), can be realized as CY-\(s\) type \(q\)-quadratic differentials. This generalizes the \(s=3\) case of [BridgelandSmith2015]. As an application, we confrim the conjectural almost Frobenis structure on the space of \(q\)-stability conditions on the Calabi-Yau-\(\mathbb{X}\) categories assoacted to type A quiver."

Ivon Dorado (Universidad Nacional de Colombia)

Let \(p\) be a prime number. A \(p\)-equipped poset is a finite poset with two kinds of points and an order relation splited into \(p\) relations. Its category of representations is equivalent to a subcategory of \(\mathcal{C}^2(proj\, \Lambda )\), where \(\Lambda\) is a finite dimensional algebra which depends on the shape of the poset, see [BautistaDorado2019].

We use the Euler form in \(\mathcal{C}^2(proj\, \Lambda )\) to asign a quadratic form to each \(p\)-equipped poset, and a coordinate vector with natural entries to each representation. If one of the vectors evaluated in the quadratic form is equal to 1 or \(p\), there is only one representation, up to isomorphism, corresponding to the vector. If it is equal to 0, there is a parametric family of representations with the same coordinate vector. If it is less than 0, then the poset is wild.

By graphing with Geogebra some polynomial equations corresponding to the quadratic forms and finding its natural solutions, we obtain information about the type of representation and we can graph the Auslander-Reiten quiver on the surfaces determined. Let me show you some examples [Dorado2020, CommunicatonEVAH]

Auslander-Reiten quiver of the 3-equipped poset !K11|72x121, 75% !imagen|690x419, 50% !imagen|690x419, 50%

Auslander-Reiten quiver of the 3-equipped poset !F20|70x130, 75% !imagen|690x405, 50%

Coordinate vectors of the representations of !2F19|67x124, 75% !imagen|690x443, 50%

The quadratic form of the poset !W10|95x58, 75% evaluated in the following points is less than 0, so the poset is wild !imagen|690x349, 50%

Graph of the quadratic form of !W10|95x58, 75% equal to -2, with some natural vectors on it !imagen|690x349, 50%

Coordinate vectors of some representations of !W10|95x58, 75%, \( \) which belong to the component of the simple projective representation in the corresponding Auslander-Reiten quiver !imagen|690x349, 50%"

A class of modules is said to satisfy the Schroeder - Bernstein property if there is an isomorphism between two modules L and M whenever there are two bijective morphisms of the form L —→ M and M —> L. We know from Bumby's paper [1] that the class of injective modules satisfy the Schroeder - Bernstein property. We refer to [3] for recent results on classes satisfying the Schroeder - Bernstein property. I summarize in the sequel the main results contained in the joint paper [2] with Derya Keskin Tutuncu. The symbol ++ will denote direct sums.

THEOREM 1 . Let U, V, W be non isomorphic modules with local endomorphism ring. The following facts hold: (a ) If there is an injective morphism L —> M and M —> L, then Add(L ++ M) does not have the Schroeder - Bernstein property. ( b ) If there is a surjective morphism V —> W, then Add(V ++ W) may have the Schroeder - Bernstein property. ( c ) If there is an injective morphism U — V ++ W, then Add(U ++ V ++ W) does not have the Schroeder - Bernstein property.

THEOREM 2. Let R be a ring, let L, M, I, Q, A, B be R-modules such that L = I ++ A and M = Q ++ B, where I and Q are injective while A and B do not have nonzero injective summands. Assume that there exist two injective morphisms L —> M and M —> L and that one of the following conditions hold: ( a ) Hom(I, B) = 0 and Hom(Q, A) = 0; ( b ) Every epimorphic image of I and Q is injective.; ( c ) R is a principal ideal domain. Then I is isomorphic to Q .

THEOREM 3. Let R be a ring admitting a sequence of pairwise non isomorphic modules X_1, X_2, X_3 ,……….. with local endomorphism ring. Assume that there exists an injective morphism X_n —> X_n+1 for any n. Let X be the direct sum of all the X_n 's . Then X does not have the Schroeder - Bernstein property.

EXAMPLE 4 . The classes of preprojective, preinjective and regular modules over the Kronecker algebra do not satisfy the Schroeder - Bernstein property.

REMARK 5. Let X be a module such that there is an injective morphism X ++ X —> X . Let A and B be submodules of X . Then there are injective morphisms X ++ A —> X ++ B and X ++ B —> X ++ A , but X ++ A and X ++ B are not necessarily isomorphic.

[ 1 ] R. T. Bumby, Modules which are isomorphic to submodules of each other, Arch. Math. 16 (1965), 184 - 185. [ 2 ] G. D'Este and D. Keskin Tutuncu, The Schroeder - Bernstein property for modules over algebras, preprint 2020 . [ 3 ] P. A. Guil Asensio, B. Kalabogaz and A. K. Srivastava, The Schroeder - Bernstein problem for modules, J. Algebra 468 (2018), 153 - 164."

Jie Pan (Zhejiang University)

Joint work with *Fang Li*

Motivated by the phenomenon that compatible Poisson structures on a cluster algebra play a key role on its quantization (that is, quantum cluster algebra), we introduce the second quantization of a quantum cluster algebra, which means the correspondence between compatible Poisson structures of the quantum cluster algebra and its secondly quantized cluster algebras.

We define a Poisson structure to be compatible on a quantum cluster algebra if it is log-canonical with every cluster, i.e., for any cluster \(\tilde{X}=(X_{1},\cdots,X_{m})\), \(\{X_{i},X_{j}\}=\omega_{ij}X^{e_{i}+e_{j}}\) holds, where \(\omega_{ij}\in\mathbb{Z}[q^{\pm \frac{1}{2}}]\) and \(i,j\in[1,m]\). By direct calculation we obtain the mutation formula of Poisson matrices corresponding to a compatible Poisson structure $

\omega_{ij}^{\prime}=
\left\{
  \begin{array}{lcr}
    q^{\frac{1}{2}(\lambda_{jk}-\sum \limits_{t=1}^{m}[b_{tk}]_{+}\lambda_{jt})}H &&if\quad i=k\neq j\\
    -\omega_{ki}^{\prime}&&if\quad j=k\neq i\\
    \omega_{ij}&&otherwise
  \end{array}
\right .

$ and then a characterization for two adjacent clusters to be log-canonical with a Poisson structure.

Theorem. If \(\tilde{X}\) is log-canonical with a Poisson structure \(\left\{-,-\right\}\) on a quantum cluster algebra \(A_q\) and \(\left\{X_{i},X_{j} \right\}=\omega_{ij}X^{e_{i}+e_{j}}\) for any \(i,j\in[1,m]\), then \(\mu_{k}(\tilde{X})\) is log-canonical with it if and only if the following conditions hold for any \(j\in[1,m],k\in[1,n],k\neq j\):

* For any \(u\in [1,m]\), if \(b_{uk}\neq0\), then \(\frac{\omega_{uj}}{\omega_{kj}}=\frac{q^{\frac{1}{2}\lambda_{u j}}-q^{\frac{1}{2}\lambda_{ju}}}{q^{\frac{1}{2}\lambda_{kj}}-q^{\frac{1}{2}\lambda_{jk}}}\).

* For any \(u,v\in [1,m]\), if \(b_{uk}b_{vk}\neq 0\), then \(\frac{\omega_{uj}}{\omega_{vj}}=\frac{q^{\frac{1}{2}\lambda_{uj}}-q^{\frac{1}{2}\lambda_{ju}}}{q^{\frac{1}{2}\lambda_{vj}}- q^{\frac{1}{2}\lambda_{jv}}}\).

* \(\sum\limits_{t:\lambda_{tj}=0}\omega_{tj}b_{tk}=0\).

Then, similar to the quantization of a cluster algebra, we define the so-called second quantization of a quantum cluster algebra associated to a compatible Poisson structure on it with quasi-exchange relations as \(Y_{t}^{e_{i}}Y_{t}^{e_{j}}=p^{W_{ij}}q^{\lambda_{ij}}Y_{t}^{e_{j}} Y_{t}^{e_{i}},\forall i,j\in[1,m]\), for a compatible triple \((\tilde{B}(t),\Lambda(t),W(t))\) and any cluster \(\tilde{Y}(t)=\left\{Y_{t}^{e_{1}},Y_{t}^{e_{2}},\cdots,Y_{t}^{e_{n}},Y^{e_{n+1}},\cdots,Y^{e_{m}}\right\}\).

\(\clubsuit\) The secondly quantized cluster algebra \(A_{p,q}(SL(2))\) of \(Fun_{\mathbb{C}}(SL_{q}(2))\) is given as a non-trivial second quantization, which may be realized as a parallel supplement to two parameters quantization of the general quantum group.

\(\clubsuit\) A class of quantum cluster algebras with coefficients is obtained which possess a non-trivial second quantization.

\(\clubsuit\) Let \(A_{q}\) be a quantum cluster algebra without coefficients. Then a Poisson structure \(\left\{-,-\right\}\) on \(A_{q}\) is compatible with \(A_{q}\) if and only if it is locally standard on \(A_{q}\). Hence the second quantization of \(A_{q}\) is always trivial."

Diaz, Yariana (University of Iowa) Gilbert, Cody (University of Iowa) Kinser, Ryan (University of Iowa)

Stability functions were introduced in [Schofield91], [King94], [Rudakov97], [Bridgeland07]. We give an efficient description of stability functions on Dynkin quivers which make all indecomposable representations of \(Q\) stable. A priori, verifying that a given stability function has this property involves checking an inequality for every pair \(W < V\) where \(V\) ranges over all indecomposable representations of \(Q\) and \(W\) ranges over all proper, nonzero subrepresentations. We use Auslander-Reiten theory to show that it suffices to check a much smaller subset of these inequalities. Total stability for type \(\mathbb{A}\) quivers was previously studied in [ApruzzeseIgusa19], [HuangHu20], [Kinser20], [BarnardGunawanMeehanSchiffler20].

Let \(\mathrm{rep}^*(Q)\) be the collection of nonzero (finite-dimensional) representations of \(Q\). Recall that a function \(\phi \colon \mathrm{rep}^*(Q) \to \mathbb{R}\) is called a *stability function* if it satisfies the *see-saw property*, meaning that for any short exact sequence of nonzero objects \[0 \to A \to B \to C \to 0\] one of the following occurs: \(\) \phi(A) < \phi(B) < \phi(C),\quad \text{or}\quad \phi(A) > \phi(B) > \phi(C),\quad \text{or}\quad \phi(A) = \phi(B) = \phi(C). \(\) A nonzero representation \(V\) is *\(\phi\)-stable* if \(\phi(W) < \phi(V)\) for all \(0 < W< V\).

Theorem. Let \(\phi\) be a stability function on a Dynkin quiver \(Q\). Then every indecomposable representation of \(Q\) is \(\phi\)-stable if and only if \(\phi(\tau V) < \phi(V)\) for all indecomposable, non-projective representations \(V\) such that the associated almost split sequence \[(\star)\quad 0 \to \tau V \to E \to V \to 0\] has indecomposable middle term \(E\). \(\square\)

The idea of the proof is to first reduce to short exact sequences where all three terms are indecomposable. We then connect these short exact sequences with those of the form \((\star)\) using a variety of methods with Auslander-Reiten quivers for each of the \(\mathbb{A}, \mathbb{D}, \mathbb{E}\) types."

Xiao-Wu Chen (USTC) Jian Liu (USTC) Ren Wang (USTC)

Background Let \(k\) be a field, and \(A, B\) be two finite dimensional \(k\)-algebras. A singular equivalence between \(A\) and \(B\), means a triangle equivalence between the singularity categories \(\mathbf{D}_{\rm sg}(A)\) and \(\mathbf{D}_{\rm sg}(B)\). Let \(M\) be an \(A\)-\(B\)-bimodule, which is finitely generated projective on each side. The functor \(M\otimes_B-\) restricts to singularity categories. The functor \({\rm Hom}_A(M, -): \mathbf{K}_{\rm ac}(A{-{\rm Inj}})\rightarrow \mathbf{K}_{\rm ac}(B{-{\rm Inj}})\) between the homotopy categories of acyclic complexes of injective modules is a triangle functor.

The question when \(M\otimes_B-\) yields a singular equivalence is of interest. It turns out that this problem is equivalent to the one when the above \({\rm Hom}_A(M, -)\) is a triangle equivalence.

Recently, it is proved in [Chen-Li-Wang2020] that Keller's conjecture for singular Hochschild cohomology is invariant under singular equivalences with levels, in the sense of [Wang2015]. Therefore, we are interested in constructing singular equivalences with levels. We mention that related results in the Gorenstein cases are obtained in [Dalezios2020].

Main Results Let \(A=kQ/I\) be a quadratic monomial algebra with \(\mathcal{R}\) its relation quiver, and \(B=k\mathcal{R}/J\) be the associated algebra with radical square zero. We construct an explicit \(A\)-\(B\)-bimodule \(M\), and obtain the following theorem and proposition.

Theorem The above functors \(M\otimes_B-\) and \({\rm Hom}_A(M, -)\) are triangle equivalences. Moreover, if \(A\) is Gorenstein, then there is a \(B\)-\(A\)-bimodule \(N\) such that \((M, N)\) defines a singular equivalence with level.

Proposition

The \(A\)-dual module \({\rm Hom}_A(M, A)\), as a left \(B\)-module, has finite projective dimension if and only if any vertex in \(\mathcal{R}\) is left-bounded provided that it is a source or has in-degree at least two. In this case, there is a \(B\)-\(A\)-bimodule \(N\) such that \((M, N)\) defines a singular equivalence with level.

Here, A vertex \(i\) in a finite quiver is said to be left-bounded, provided that there is a uniform bound of all the paths starting at \(i\). The in-degree of a vertex \(i\) is defined to be the cardinality of the set of all arrows ending at \(i\).

Examples We give examples of the algebra \(A\) such that there is a singular equivalence with level between \(A\) and its associated algebra of radical square zero. 1. (Gorenstein cases.) Gentle algebras. 2. (Non-Gorenstein case.) Let \(A\) be given by the following quiver \(Q\):

      ![[upload://uDGj5yFAWSDw8wyyQUoiVuQ7jBl.jpeg|quiver|394x127, 50%]] 
  with relations given by \(\mathbf{F}=\{\alpha\beta,\beta\gamma,\gamma\gamma\}\)."

Sonia Trepode - On representation dimension three algebras and smallest Auslander generators, Sonia Trepode (Universidad Nacional de Mar del Plata)

Joint work with Edson Ribeiro Alvares, Clezio Braga and Heily Wagner.

We consider the smallest generator-cogenerator module which consists of the direct sum of all indecomposable projective modules and all indecomposable injective modules. We consider the class of algebras such that the endomorphism algebra of the smallest generator-cogenerator has global dimension equal to three. We call these algebras *representation hereditary algebras*. In particular, if the algebra is representation infinite then the representation dimension is equal to three and the Auslander generator is the smallest generator-cogenerator module.

We prove our first theorem.

Theorem 1. If \(A\) is a hereditary representation algebra then \(A\) is torsion-less finite.

Under the hypothesis that there are not morphisms from an injective module to a projective module we prove the following theorem.

Theorem 2. Let \(A\) be a representation hereditary algebra such that \(Hom_A(DA,A) = 0\), then \(A\) is quasi-tilted. Furthermore, if \(A\) is a representation infinite algebra, then \(A\) is a concealed algebra.

We show an example of a representation finite algebra \(A\) satisfying \(Hom_A(DA,A) \not = 0\), which is a representation hereditary algebra and is not a quasi-tilted algebra.

We study radical square zero algebras.

Theorem 3. Let \(A\) be a non hereditary radical square zero algebra. Then \(A\) is a representation hereditary algebra if and only if each connected component of the separated quiver \(\bar Q\) is of \(\mathbb{A}_1\), \(\mathbb{A}_2\) or \(\mathbb{A}_3\) type and, if any component is \(a \to b'\), then \(a\) is a source in \(Q\) or \(b\) is a sink in \(Q\)."

[Qiuning Du] (Zhejiang University)

Joint work with *Fang Li and Jie Pan*

Cluster quiver is a significant notion in the theory of cluster algebras. The algebraic and combinatorial properties are concerned in the study of cluster quivers. As the combinatorial properties of quivers, there are two aspects: one is the number of vertices, the other one is the number of arrows. Mutation preserves the number of vertices of cluster quivers, which is corresponding to the rank of cluster algebras. So then we need to consider how the number of arrows is distributed in a mutation equivalence class.

Let \(Q\) be a cluster quiver, \(t\) be the number of arrows of \(Q\). We call \(W=\{t|t=|Q'_1|, where\ Q'\in Mut[Q]\}\) the distribution set of numbers of arrows for \(\mathbf{Mut[Q]}\). When \(Q\) arises from a surface \(\mathbf{S}\), \(W\) is called the distribution set of numbers of arrows for the surface \(\mathbf{S}\). Let \(t_{max}\) (respectively, \(t_{min}\)) be the maximum (respectively, minimum) in the distribution set \(W\). The number of arrows for \(Mut[Q]\) is said to present a continuous distribution if \(W=[t_{min},t_{max}]\cap \mathbb{N}\), where \([t_{min},t_{max}]\) is an interval (the situation that \(t_{min}=t_{max}\) is allowed).

We call a path a complete path for \(\mathbf{S}\) in the graph \(\mathbf{E^\circ(S,M)}\) if vertices on this path are different and the set of numbers of arrows of quivers associated to triangulations on this path is an interval in \(\mathbb{N}\) from \(t_{min}\) to \(t_{max}\).

A cluster quiver \(Q\) includes those arrows between mutable vertices and frozen vertices. In order to emphasize this view, we call also \(Q\) an extended cluster quiver. Sometimes we ignore the arrows between mutable vertices and frozen vertices, that is, we only consider the full sub-quiver of \(Q\) consisting of mutable vertices, denoted as \(\check{Q}\), which is called the exchange cluster quiver of \(Q\).

Theorem 1. The maximum and minimum numbers \(t_{max}\) (or \(\check{t}_{max}\)) and \(t_{min}\) (or \(\check{t}_{min}\)) of arrows for \(Mut[Q]\) (or \(Mut[\check{Q}]\)) are obtained respectively for \(Q\) (or \(\check{Q}\)) from a surface \(\mathbf{S}\).

Theorem 2. When the triangulation is associated with the exchange cluster quiver (respectively, extended cluster quiver), there is a complete path in the graph \(\mathbf{E^\circ(S,M)}\) of a surface \(\mathbf{S}\) which is not a twice-punctured digon and a forth-punctured sphere (respectively, an once-punctured triangle and a forth-punctured sphere).

Theorem 3. Let \(\check Q\) (resp. \(Q\)) be an exchange (resp. extended) cluster quiver of finite mutation type, then the distribution set of numbers of arrows for \(Mut[\check Q]\) (resp. \(Mut[Q]\)) presents a continuous distribution unless the exceptional cases (1) and (2) (resp. (3) and (4)) as follows: (1) the cluster quiver arises from a twice-punctured digon or a forth-punctured sphere, (2) the cluster quiver is of type \(X_6\) or \(X_7\), (3) the cluster quiver arises from a once-punctured triangle or a forth-punctured sphere, (4) the cluster quiver is of type \(X_6\) or \(X_7\).

\(\spadesuit\) Moreover, for a cluster quiver \(Q\), we find that in case of infinite mutation type, the number of arrows for \(Mut[Q]\) does not present a continuous distribution." Fang Li - On topological representation of a quiver, [Fang Li] (affiliation)

Let \(S\) be a semigroup and \(M\) a non-empty topological space. If the map \(L_S: S\times M\longrightarrow M\) satisfies \(L_{s_{2}s_{1}}(m)=L_{s_{2}}\big(L_{s_{2}}(m)\big)\) %\(\varphi(s_{2},\varphi(s_{1},m))=\varphi(s_{2}s_{1},m)\), $\forall s_{1},s_{2}\in S\(, \)\forall m\in M\(, then \)(M, L_S)$ is called a {\bf left \(S\)-topological system}. Let \(M,N\) are two \(S\)-Systems of topological spaces, a continuous map \(f:M\longrightarrow N\) is called an {\bf \(S\)-morphism} from \(M\) to \(N\), if \(f(sm)=sf(m)\), $\forall s\in S\( and \)\forall m\in M$. All left \(S\)-topological systems and all \(S\)-morphisms between them constitute a category, denoted by \(S\)-\(\mathcal{TOP}\).

Let \(\Gamma=(\Gamma_{0},\Gamma_{1})\) be a quiver with \(\Gamma_{0}\) the set of vertices and \(\Gamma_{1}\) the set of arrows between vertices. A {\bf topological representation} \((T,f)\) of a quiver \(\Gamma=(\Gamma_{0},\Gamma_{1})\) is a family of pairs of topological space $\{T_{i}: i\in\Gamma_{0}\}\( together with continuous map \)f_{\alpha}$: \(T_{i}\rightarrow T_{j}\) for each arrow \(\alpha\): \(i\rightarrow j\). Let \((T, f)\) and \((T', f')\) be two topological representations of \(\Gamma\). A {\em morphism} \(h\): \((T,f)\rightarrow(T',f')\) between two top-representations of \(\Gamma\) is a collection of continuous maps \(\{h_{i}:T_{i}\rightarrow T'_{i}\}_{i\in\Gamma_{0}}\) such that for each arrow \(\alpha\): \(i\rightarrow j\) in \(\Gamma_{1}\) the following diagram:

\(\) \xymatrix{ &T_i\ar[r]^{h_i}\ar[d]_{f_{\alpha}}&T_i^{'}\ar[d]^{f_{\alpha}^{'}}
&T_j\ar[r]^{h_j}&T_j^{'} } \(\)"

"Relative homological algebra \& algebraic right triangulated categories

Aran Tattar (University of Leicester)

We show that, by using notions from relative homological algebra, one can construct Quillen exact structures on an additive category. We characterise when these exact structures have a 'one-sided Frobenius' property and show that then the stable category has has an `algebraic' right triangulated structure. Furthermore, we prove that these categories are extriangulated and all aisles of t-structures in an algebraic triangulated category arise in this way. This is based on current work-in-progress.

Throughout let \(\mathcal{C}\) be an additive category and let \(\mathcal{D} \subset \mathcal{C}\) be a subcategory.

We recall that a morphism \(f: X \to Y\) in \(\mathcal{C}\) is \(\mathcal{D}\)*-monic* if \(\mathcal{C}(f, D): \mathcal{C}(X,D) \to \mathcal{C}(Y,D)\) is surjective for all \(D \in \mathcal{D}\). The idea is that, by restricting our attention to \(\mathcal{D}\)-monic morphisms, we may treat the objects of \(\mathcal{D}\) as injective. An important example of \(\mathcal{D}\)-monic morphisms are left-\(\mathcal{D}\)-approximations.

This allows us to construct new exact structures.

Proposition 1. Let \(\mathcal{E}\) be a Quillen exact structure on \(\mathcal{C}\) and let \(\mathcal{E}_{\mathcal{D}}\) be the class of all conflations in \(\mathcal{E}\) such that the inflation is \(\mathcal{D}\)-monic. Then \(\mathcal{E}_{\mathcal{D}}\) is an exact structure on \(\mathcal{C}\) and the objects of \(\mathcal{D}\) are injective with respect to this exact structure. Furthermore, there are conditions (A) on the structure of \(\mathcal{D}\) for \(\mathcal{E}_\mathcal{D}\) to have enough injectives.

When \(\mathcal{D}\) satisfies the conditions (A), \( \mathcal{E}_\mathcal{D}\) has a `one-sided Frobenius' property. Thus, by combining the above statement with its dual gives a characterisation of Frobenius exact categories ([Happel1988])) in terms of relative homological algebra. This also leads us to study the structure of the stable category, \(\underline{\mathcal{C}}_\mathcal{D}\), which now has a nice right triangulated structure. We recall that the shift functor of a right triangulated category is said to be a *right-semi equivalence* if it is full, faithful and if its image is closed under extensions [AssemBeligiannisMarmaridis1998].

Theorem 1. Suppose that \(\mathcal{D}\) satsifies the conditions (A). Then \(\underline{\mathcal{C}}_\mathcal{D}\) is a right triangulated category with right-semi equivalence shift with right triangles coming from conflations in \(\mathcal{E}_{\mathcal{D}}\). In this case, we call \(\underline{\mathcal{C}}_\mathcal{D}\) an *algebraic right triangulated category*.

We make the observation that right triangulated categories with right-semi equivalence shift are extriangulated in the sense of [NakaokaPalu2019] and thus provide another class of extriangulated categories that are neither exact nor triangulated.

Examples of subcategories \(\mathcal{D}\) that result in algebraic right triangulated categories come from cotorsion pairs. We also show that all t-structures in an algebraic triangulated category arise in this way. 

Theorem 2. Every aisle of a t-structure in an algebraic triangulated category is the shift of an algebraic right triangulated category."

Sebastian Eckert (Bielefeld University) *This is part of the author's doctorate studies under the supervision of W. Crawley-Boevey* $\def\eps{\varepsilon} \DeclareMathOperator{\mods}{mod} \def\isom{\mathrel{\cong}} \DeclareMathOperator{\coker}{coker}$ The notion of amenable representation type was introduced by [Elek2017]. Roughly speaking, a finite dimensional algebra is of amenable type if for all \(\eps > 0\), every finite dimensional module has a submodule which is a direct sum of modules which are small with respect to \(\eps\) such that the quotient is also small in that respect. Subfamilies of modules having this property are then called hyperfinite. Elek conjectures that finite dimensional algebras are tame iff they are amenable and shows that string algebras are amenable.

Definition. Let \(k\) be a field, \(A\) be a finite dimensional \(k\)-algebra and let \(\mathcal{M} \subseteq \mods A\) be a family of finite dimensional \(A\)-modules. Now \(\mathcal{M}\) is called weakly hyperfinite provided for every \(\eps>0\) there exists some \(L_\eps> 0\) such that for every \(M \in \mathcal{M}\) there is a homomorphism \(\theta \colon N \to M\) for some \(N \in \mods A\) such that
\(\)
\dim_k \ker \theta \leq \eps\dim M, \quad \dim_k \coker \theta \leq \eps\dim M,
\(\)
and there are modules \(N_1,\dots,N_t \in \mods A\) with \(\dim_k N_i \leq L_\eps\) such that \(N \isom \bigoplus_{i=1}^{t}N_i\).
A \(k\)-algebra \(A\) is said to be of weakly amenable representation type provided \(\mods A\) itself is a weakly hyperfinite family.

Continuing to work towards the conjecture, which draws similarities to the tame-wild dichotomy, we have focussed on finding explicit, non-hyperfinite families for wild Kronecker algebras. As it turns out, dimension expanders are useful for this.

Definition. Let \(k\) be a field, \(d \in \mathbb{N}\), \(0 < \eta \leq 1\) and \(\alpha > 0\). Given a vector space \(V\) and a set \(\{T_1,\dots T_d\}\) of endomorphisms of \(V\), the pair \((V,\{T_i\}_{i=1}^{d})\) is called an \((\eta,\alpha)\)-dimension quasi-expander of degree \(d\) provided for every subspace \(W \subset V\) of dimension at most \(\eta \dim_k V\), we have that
\(\)

\dim_k \sum_{i=1}^d T_i(W) \geq (1+\alpha) \dim_k W.

\(\)

A sequence of dimension quasi-expanders of degree \(d\) of increasing dimension gives rise to a non-weakly hyperfinite family for the \(d\)-Kronecker algebra \(k\Theta(d)\):

Proposition. Let \(k\) be a field, \(d \in \mathbb{N}\) and \(\eta, \alpha > 0\). If \(\{(V_i,\{T_l^{(i)}\}_{l=1}^{d})\}_{i \in I}\) is a sequence of \((\eta,\alpha)\)-dimension quasi-expanders of degree \(d\) such that \(\dim V_i\) is unbounded, then the induced sequence of \(k\Theta(d)\)-modules \(\left\{V_i \stackrel{\{T_l^{(i)}\}}{\substack{\rightarrow \\ \tiny\vdots\\ \rightarrow}} V_i\right\}_{i \in I}\) is not weakly hyperfinite.

By a result of [LubotzkyZelmanov2008], there is a variety of \((\frac{1}{2},\alpha)\)-dimension quasi-expanders of degree three over the complex numbers, while results of [DvirWigderson2010] and [BourgainYehudayoff2013] allow for the construction of degree-\(d\) dimension quasi-expanders over any field.

Theorem. Let \(k\) be any field. Then there exists some \(d\geq3\) such that the wild Kronecker algebra \(k\Theta(d)\) is not of (weakly) amenable representation type.

We use the related notion of fragmentability from graph theory (see [EdwardsMcDiarmid1994]) to show that the preprojective and preinjective components of the generalized Kronecker quiver algebras form hyperfinite families, using the description of tree modules by [Ringel199810046-5)].

Finally, we identify functors that preserve non-weakly hyperfinite families. The interpretation functors for finitely controlled wild algebras of [GregoryPrest2016] then turn out to be sufficient to prove Elek's conjecture for controlled wild algebras.

Theorem. Let \(k\) be an algebraically closed field. If \(A\) is a finitely controlled wild algebra, then \(A\) is not of amenable representation type."