Seminars in Algebra

Tuesday, August 22nd, 2023, at 13:15 in room 656 (Simastuen), SBII

Speaker: Darius Dramburg (Uppsala University)

Title: Classifying 2-representation infinite algebras of type Ã

Abstract: n-representation infinite algebras play an important role in higher AR-theory. One of their features is that their higher preprojective algebras are (n+1)-Calabi-Yau, and higher type à was defined by Herschend-Iyama-Oppermann in terms of the higher preprojective algebra. We therefore need to understand the preprojective algebras when trying to classify n-representation infinite algebras. In this talk, I will begin by giving a brief overview of n-representation infinite algebras, their preprojective algebras, and how they fit into higher AR-theory. Next, we introduce some skew-group algebras of polynomial rings by finite subgroups G < SL_{n+1}, which are (n+1)-Calabi-Yau. However, not all of them are higher preprojective, and we will discuss the (in general open) problem of deciding which ones are. Setting n=2 and G abelian, I will then present a classification of the 3-preprojective algebras of type Ã, and explain how this leads to a classification of all 2-representation infinite algebras up to a certain kind of mutation. Time permitting, I will explain the challenge when generalising to n>2, and how our results relate to the structure of (abelian) Gorenstein quotient 3-folds.

This is based on joint work with Oleksandra Gasanova.

Tuesday, August 29th, 2023, at 13:10 in room F2, Gamle fysikk

Speaker: Erlend D. Børve (Institut Fourier)

Title: τ-tilting finite incidence algebra of posets

Abstract: Let A be a finite-dimensional algebra with n isoclasses of simples. To an indecomposable module (or shifted indecomposable projective) M, we can associate a certain rational function u_M in Q(y_1,…y_n). In the case that A is the path algebra of a Dynkin quiver, the u_M are very closely related to a subset of the X-cluster variables, in the sense of Fock and Goncharov. Using a result of Domínguez and Geiss, we show that, when A is of finite representation type, the u_M satisfy some beautiful relations. When A is not of finite type, I will discuss analogues of the finite representation type relations which we expect to hold in the ring of formal power series. We are able to establish these relations in the case of gentle algebras. I will attempt to say a little something about the motivation for this problem, which is drawn from the physics of scattering amplitudes. This is joint work with Nima Arkani-Hamed, Hadleigh Frost, Pierre-Guy Plamondon, and Giulio Salvatori.

Tuesday, October 10th, 2023, at 13:10 in room F2, Gamle fysikk

Speaker: Carlo Klapproth (Aarhus University)

Title: Obstructions to loops and 2-cycles in the quivers of categories (jt. w. Martin Kalck and Nebojsa Pavic)

Abstract: The no-loop conjecture states that a finite dimensional algebra kQ/I of finite global dimension cannot have loops in its quiver Q. A generalisation of this conjecture has been shown by Igusa—Liu—Paquette in 2011. We use this result and the theory of functor categories to show that certain categories cannot have loops in their Auslander—Reiten quiver. This result applies for example to the singularity categories of finite dimensional Iwanaga—Gorenstein algebras. We also present a generalisation of a result by Buan—Iyama—Reiten—Scott giving us an obstruction for 2-cycles in the Auslander—Reiten quiver of certain categories and discuss our motivating application in algebraic geometry.

Tuesday, November 21st, 2023, at 13:10 in room F2, Gamle fysikk

Speaker: Rudradip Biswas (University of Warwick)

Title: Generalizing Neeman's work on bounded t-structures on derived perfect categories and derived bounded categories

Abstract: I will present some new work (joint with Hongxing Chen, Kabeer Manali Rahul, Chris Parker, and Junhua Zheng) where we generalise Neeman's work by showing that if we take a triangulated category with arbitrary coproducts admitting a single compact generator, and assume that the opposite category of the compacts has finite finitistic dimension (this notion is new from our paper), then the compacts coincide with their completion with respect to a good metric à la Neeman. Neeman had worked with the derived category of quasi-coherent complexes over sufficiently good schemes as the big triangulated category. Our result is a major abstract generalisation of Neeman's because one can "complete" the category of perfect complexes to get the derived bounded category. Neeman also proved that under some reasonably restrictive conditions on the scheme, all bounded t-structures on the derived bounded category of coherent complexes over the scheme are equivalent. This result has also been majorly generalised by us as we have succeeded in removing those restrictive conditions.

Monday, January 23rd, 2023, at 11:15 in room 656 (Simastuen), SBII

Speaker: Henning Krause (Bielefeld University)

Title: Central support for triangulated categories

Abstract: Various notions of support have been studied in representation theory (by Carlson, Snashall-Solberg, Balmer, Benson-Iyengar-Krause, Friedlander-Pevtsova, Nakano-Vashaw-Yakimov, to name only few). My talk offers some new and unifying perspective: For any essentially small triangulated category the centre of its lattice of thick subcategories is introduced; it is a spatial frame and yields a notion of central support. A relative version of this centre recovers the support theory for tensor triangulated categories and provides a universal notion of cohomological support. Along the way we establish Mayer-Vietoris sequences for pairs of central subcategories.

Monday, January 23rd, 2023, at 12:15 in room 656 (Simastuen), SBII

Speaker: Hugh Thomas (Université du Québec à Montréal)

Title: An analogue of X-cluster variables for finite-dimensional algebras

Abstract: Let A be a finite-dimensional algebra with n isoclasses of simples. To an indecomposable module (or shifted indecomposable projective) M, we can associate a certain rational function u_M in Q(y_1,…y_n). In the case that A is the path algebra of a Dynkin quiver, the u_M are very closely related to a subset of the X-cluster variables, in the sense of Fock and Goncharov. Using a result of Domínguez and Geiss, we show that, when A is of finite representation type, the u_M satisfy some beautiful relations. When A is not of finite type, I will discuss analogues of the finite representation type relations which we expect to hold in the ring of formal power series. We are able to establish these relations in the case of gentle algebras. I will attempt to say a little something about the motivation for this problem, which is drawn from the physics of scattering amplitudes. This is joint work with Nima Arkani-Hamed, Hadleigh Frost, Pierre-Guy Plamondon, and Giulio Salvatori.

Wednesday, February 9th, 2023, at 14:15 in room 656 (Simastuen), SBII

Speaker: Amit Shah (Aarhus University)

Title: Characterising Jordan-Hölder extriangulated categories via Grothendieck monoids

Abstract: The notions of composition series and length are well-behaved in the context of abelian categories. And, in addition, each abelian category satisfies the so-called Jordan-Hölder property/theorem. Unfortunately, these ideas are poorly behaved for triangulated categories. However, with the introduction of extriangulated categories, it is interesting (at least for me and my collaborators Thomas Brüstle, Souheila Hassoun and Aran Tattar) to see what sense we can make of these concepts for extriangulated categories.

I’ll present a result that characterises Jordan-Hölder, length extriangulated categories using the Grothendieck monoid of an extriangulated category. This is motivated by the exact category setting as considered by Enomoto, in which it becomes apparent that Grothendieck monoid is more appropriate to look at than the Grothendieck group. I’ll present some examples coming from stratifying systems. In fact, developing stratifying systems for extriangulated categories was the original motivation of our article.

Wednesday, February 15th, 2023, at 14:15 in room 656 (Simastuen), SBII

Speaker: Raphael Bennett-Tennenhaus (Aarhus University)

Title: Corner replacement for Morita contexts

Abstract: A celebrated result of Morita characterises equivalences between categories of modules over unital rings. Bass exposed these results in terms of a more general situation, known today as a Morita context. Since then various authors have extended the setting of so-called Morita theory to cater for rings which need not be unital. For example, Ánh and Márki developed the theory for unital modules over rings with local units.

I will begin by explaining how Morita contexts with a common ring can be composed, excised and ligated back together. Specifying, I will then highlight the way in which replacing a corner subring is compatible with Morita equivalences (which I suspect was `folklore’), and give applications to representations of species. Time permitted, I will consider a potential connection to locally finitely presented additive categories. This talk is based on a recent arxiv article 2301.09518.