# Seminars in Algebra

## Fall 2015

## November 24th 2015, 14:15 - 15:15 in 656

**Title:** The Lie bracket in Hochschild cohomology via the homotopy category of projective bimodules (II)

**Speaker:** Reiner Hermann

**Abstract:** This talk aims for employing the results that have been presented in the preceding one in order to realise the Lie bracket in terms of the (bounded below) homotopy category of projective bimodules. After recalling Schwede’s loop bracket construction, I will explain how one can take advantage of the tensor triangulated structure of the homotopy category of projective bimodules to produce, for two given graded endomorphisms of the tensor unit, an element of a fundamental group of a certain morphism. I will conclude by sketching how Buchweitz’ and Schwede’s results can be combined to verify that this construction indeed gives rise to the Lie bracket in Hochschild cohomology. Thus, morally, the existence of the Lie bracket in Hochschild cohomology can be regarded as a shadow of the (often bemoaned) fact that taking cones in a triangulated category is non-functorial in general.

## November 17th 2015, 14:15 - 15:15 in 656

**Title:** The Lie bracket in Hochschild cohomology via the homotopy category of projective bimodules (I)

**Speaker:** Johan Steen

**Abstract:** This is the first of two talks, based on joint work with Reiner Hermann, where we interpret the Lie algebra structure of the Hochschild cohomology of an (associative) algebra inside the bounded below homotopy category of projective bimodules. The aim for the first talk is to describe the work of others that led us to this project. Specifically, I will discuss the fundamental group of a morphism in a triangulated category (due to Buchweitz), the so-called Retakh isomorphism, which relates an extension group to a certain fundamental group, and furthermore show that this can be recovered from the work of Buchweitz in a very natural way.

## November 11th 2015, 13:15 - 14:15 in 734

**Title:** Join-irreducibles of weak order on finite Coxeter groups

**Speaker:** Hugh Thomas

**Abstract:** I will explain weak order on Coxeter groups, and then discuss several approaches to understanding one of its key features: its join-irreducible elements. I will discuss recent work with Osamu Iyama, Nathan Reading and Idun Reiten, in which we construct a bijection between join-irreducible elements and a certain class of modules over the preprojective algebra. I will also discuss older work of Reading, which gives a more combinatorial/geometric approach to the same topic. Finally, if there is time, I will discuss an ongoing project with David Speyer in which we relate the two approaches.

## November 10th 2015, 14:15 - 15:15 in 656

**Title:** Semi-invariant pictures, c-vectors, maximal green sequences

**Speaker:** Gordana Todorov

**Abstract:** See the link abstract

## November 3rd 2015, 14:15 - 15:15 in 656

**Title:** Orbit categories and self-injective algebras

**Speaker:** Karin Marie Jacobsen

**Abstract:** It is easy to see that there is a strong similarity between the AR-quivers of orbit categories and stable module categories of self-injective algebras. In 2013 Holm and Jørgensen classified all cluster categories that are equivalent to stable module categories. Using a theorem by Amiot, we give a classification of the orbit categories that are triangulated equivalent to stable module categories. This is joint work with Benedikte Grimeland.

## October 28th 2015, 13:15 - 14:15 in 734

**Title:** Dehn twist groups as fundamental groups of spaces of (signed) quadratic differentials

**Speaker:** Yu Qiu

**Abstract:** We study the space Quad of (signed) quadratic differentials on a marked surface S. By constructing King-Qiu's twisted surface Sigma of S, we show that pi_1 Quad is isomorphic to a subgroup DTG of the mapping class group of Sigma generated by Dehn twists. We also conjecture that DTG is isomorphic to the spherical twist group for the corresponding 3-Calabi-Yau category. This conjecture has been proven by my previous work in the case when S is unpunctured and by my joint work with Jon Woolf in the case when S is a once-punctured disk (type D).

## October 21st 2015, 13:15 - 14:15 in 734

**Title:** Stability conditions on the CY completion of a formal parameter and twisted periods

**Speaker:** Akishi Ikeda

**Abstract:** In the second talk, we define a variant of a Bridgeland stability condition on the derived category of the Calabi-Yau completion for the formal parameter which is defined in the first talk. The space of these stability conditions becomes a complex manifold and admits the action of complex numbers and autoequivalences. In particular, the action of a complex number \(s\) coincides with the action of the Serre functor which is given by the shift of the formal parameter. We also discuss the relationship between central charges of these stability conditions and twisted periods of the Frobenius manifold.

## October 20th 2015, 14:15 - 15:15 in 656

**Title:** Calabi-Yau completion for a formal parameter

**Speaker:** Akishi Ikeda

**Abstract:** The aim of this work is to construct stability conditions on \(s\)-Calabi-Yau categories for a complex number \(s\). In the first talk, we introduce the notion of a Calabi-Yau completion for a formal parameter as an analogue of Keller's Calabi-Yau completion for an integer. The derived category of the CY completion for a formal parameter has the extra degree shift for the direction of the formal parameter and this shift becomes the Serre functor. As an application, we give the categorification of the representation of the Iwahori-Hecke algebra on the (deformed) root lattice associated to an acyclic quiver.

## October 14th 2015, 14:15 - 15:15 in 734

**Title:** Catalan numbers, Polytopes and the Number of Tilting Modules

**Speaker:** Lutz Hille

**Abstract:** There is an extensive study of Catalan numbers in combinatorics. Some of this is closely connected to the number of tilting modules for quivers of type A. Since the number of tilting modules is independent of the orientation for any Dynkin quiver, the number of tilting modules should be defined already in terms of the root system. It turns out, that there is a polytop, the so-called root polytop, and its volume is closely related to the number of tilting modules. In the talk we review some of the Catalan combinatorics, introduce the root polytopes and relate them to the counting problem. It turns out, that we can also count the number of strongly exceptional sequences with this method. At the end we give a short view to the corresponding problem for tame and wild quivers.

## October 14th 2015, 13:10 - 14:10 in 734

**Title:** Lifting problems concerning actions of finite groups on curves

**Speaker:** Ted Chinburg

**Abstract:** A famous problem in characteristic p algebraic geometry is to determine when the action of a finite group G on a smooth projective curve over a field k of positive characteristic p can be lifted to characteristic 0. A lift is a smooth curve with G-action over a DVR of characteristic 0 and residue field k which reduces to the original curve with group action over k. The Oort conjecture, proved recently by Obus, Wewers and Pop, shows that if G is cyclic such a lift always exists. The main problem in the subject now is whether lifts always exist when G is dihedral of order 2 p^n. I will discuss some obstructions to lifting which may help settle this problem. One new obstruction arises from the Galois structure of holomorphic differentials discussed in Frauke Bleher's talk.

## October 13th 2015, 14:15 - 15:15 in 656

**Title:** Galois structure of holomorphic differentials

**Speaker:** Frauke Bleher

**Abstract:** This talk is about joint work with Ted Chinburg and Aristides Kontogeorgis. Let X be a smooth projective curve over an algebraically closed field k of positive characteristic p. Suppose G is a finite group with non-trivial cyclic Sylow p-subgroups acting faithfully on X. We determine the kG-module structure of the module H^0(Omega_X) of holomorphic differentials of X in terms of the so-called Boseck invariants of the ramification of the action of G on X. The proof uses modular representation theory to reduce to the case in which G is a semidirect product of a normal cyclic p-subgroup and a cyclic prime-to-p group. In this case, one compares the radical filtration of H^0(Omega_X) to the global sections of subquotients of the radical filtration of the sheaf Omega_X.

## October 6th 2015, 14:15 - 15:15 in 656

**Title:** Singular equivalence and the (Fg) condition

**Speaker:** Øystein Ingmar Skartsæterhagen

**Abstract:** We show that singular equivalences of Morita type with level between finite-dimensional Gorenstein algebras over a field preserve the (Fg) condition.

## September 30th 2015, 13:15 - 14:15 in 734

**Title:** Multiserial, special multiserial, and Brauer configuration algebras

**Speaker:** Edward L. Green

**Abstract:** In joint work with Sibylle Schroll, we generalize biserial, special biserial and Brauer graph algebras. We prove various interconnections between these algebras and classify radical cubed 0 symmetric algebras in this setting.

## September 29th 2015, 14:15 - 15:15 in 656

**Title:** On the hearts of cotorsion pairs on triangulated categories

**Speaker:** Hiroyuki Nakaoka

**Abstract:** I will introduce the construction of the heart of a (twin) cotorsion pair on a triangulated category, which generalizes the heart of a t-structure and the ideal quotient by a cluster tilting subcategory. I will also refer to conditions under which the heart becomes a module category.

## September 23rd 2015, 13:15 - 14:15 in 734

**Title:** A Swiss Cheese theorem for linear operators with two invariant subspaces

**Speaker:** Markus Schmidmeier

**Abstract:** In this talk we discuss a joint project with Audrey Moore on the possible dimension types of linear operators with two invariant subspaces. Formally, we consider systems \((V, T, U_1, U_2)\) where \(V\) is a finite dimensional vector space, \(T: V\to V\) a nilpotent linear operator, and \(U_1\), \(U_2\) subspaces of \(V\) which are contained in each other and which are invariant under the action of \(T\). To each system we can associate as dimension type the triple \((dim U_1, dim U_2/U_1, dim V/U_2)\). Such systems occur in the theory of linear time-invariant dynamical systems where the subquotient \(U_2/U_1\) is used to reduce the dynamical system to one which is completely controllable and completely observable. No gaps but holes: The well-known No-Gap Theorem by Bongartz states that for a finite dimensional algebra over an algebraically closed field, whenever there is an indecomposable module of length \(n>1\), then there is one of length \(n-1\). By comparison, consider the topological space given by the dimension types of indecomposable systems in the situation where \(T\) acts with nilpotency index at most 4. Our main result states that in this space there are triples, for example \((3,1,3)\), which can not be realized as the dimension type of an indecomposable object, while each neighbor can.

## August 25th 2015, 14:15 - 15:15 in 656

**Title:** Cohomological support and tensor products

**Speaker:** William Sanders

**Abstract:** The theory of cohomological supports encodes important homological information of a module into a geometric object. In this talk, we consider cohomological supports over complete intersection rings. In particular, we investigate the cohomological support of the tensor product of two modules using the geometry of the cohomological supports of the original modules. Furthermore, we pose questions regarding the asymptotic behavior of the cohomological supports of Tor modules.

## Spring 2015

## June 24th 2015, 13:15 - 14:15 in 734

**Title:** Abelian quotients of triangulated categories

**Speaker:** Benedikte Grimeland

**Abstract:** We study abelian quotient categories A=T/J, where T is a triangulated category and J is an ideal of T. Under the assumption that the quotient functor is cohomological we show that it is representable and give an explicit description of the functor. We give technical criteria for when a representable functor is a quotient functor, and a criterion for when J gives rise to a cluster-tilting subcategory of T. We show that the quotient functor preserves the AR-structure. As an application we show that if T is a finite 2-Calabi-Yau category, then with very few exceptions J is a cluster-tilting subcategory of T.

## June 17th 2015, 13:15 - 14:15 in 734

**Title:** Extensions in gentle algebras

**Speaker:** Sibylle Schroll

**Abstract:** The aim of this talk is to give a basis for the extensions between indecomposable modules over gentle algebras in terms of string combinatorics. We will start by examining the case of gentle algebras arising as Jacobian algebras associated to triangulations of bounded unpunctured surfaces. In this case we work in the associated cluster category. We will see that the Ptolemy relations of arcs always give rise to triangles in the cluster category but that they do not necessarily give rise to extensions in the gentle algebra. In the general case of a gentle algebra, we will use its derived category to give an upper bound on the dimensions of the Ext spaces by developing a mapping cone calculus of homotopy strings. This a report on joint work in progress with Ilke Canakci and David Paukzstello.

## June 10th 2015, 13:15 - 14:15 in 734

**Title:** Frobenius d-exact categories

**Speaker:** Gustavo Jasso

**Abstract:** Frobenius d-exact categories, as their name suggests are higher analogs of Frobenius exact categories. They were introduced in order to enhance Geiss-Keller-Oppermann's (d+2)-angulated categories. In this talk I will introduce Frobenius d-exact categories, explain what is known about them and give examples of such categories occurring in nature.

## March 18th 2015, 13:15 - 14:15 in 734

**Title:** (Co)Silting, (Co)tilting, t-structures and derived equivalences

**Speaker:** Jorge Vitoria

**Abstract:** While derived equivalences between categories of modules over a ring correspond to compact tilting complexes, little is known about derived equivalences between more general abelian categories. With a suitable notion of a large (i.e., not necessarily compact) tilting object, one can establish derived equivalences between certain abelian categories. For this purpose, rather than studying the endomorphism ring of such an object, one should focus on the heart of its associated t-structure. Analogous equivalences can be obtained from large cotilting objects. In this talk, we explain this point of view on tilting theory, starting from the more general concept of large silting and cosilting objects in the derived category of a Grothendieck category. This is a report on ongoing joint work with Chrysostomos Psaroudakis.

## February 18th 2015, 13:15 - 14:15 in 734

**Title:** Gorenstein-projective modules over trivial extension algebras

**Speaker:** Chrysostomos Psaroudakis

**Abstract:** Gorenstein-projective modules over any (non-commutative) ring were introduced by Enochs-Jenda, generalizing the notion of Gorenstein-dimension zero finitely generated modules over noetherian rings due to Auslander. It is known that the homological study of the category of Gorenstein-projective modules reflects properties of the ring itself, and therefore it is an interesting problem to describe this class of modules. In this talk we show how to construct Gorenstein-projective modules over a class of trivial extension rings arising from Morita contexts. This is joint work with Nan Gao.

## February 11th 2015, 13:15 - 14:15 in 734

**Title:** Derived invariance of support varieties

**Speaker:** Øystein Ingmar Skartsæterhagen

**Abstract:** This talk is based on joint work with Julian Külshammer (Stuttgart) and Chrysostomos Psaroudakis. Support varieties for modules over finite-dimensional algebras were introduced by Snashall and Solberg in 2004, as a generalisation of the theory of support varieties for modules over group algebras. More generally, one can define the support variety of any bounded complex of modules over a finite-dimensional algebra. If a finite-dimensional algebra A satisfies a set of conditions called (Fg), then many of the results known to hold for support varieties in the case of group algebras also hold for the algebra A. We show that the (Fg) condition is invariant under derived equivalence of algebras. Moreover, we show that a derived equivalence of standard type between two finite-dimensional algebras preserves support varieties of bounded complexes.

## February 4th 2015, 13:15 - 14:15 in 734

**Title:** The quasihereditary structure of a Schur-like algebra

**Speaker:** Teresa Conde

**Abstract:** Given an arbitrary algebra \(A\), we may associate to it a special endomorphism algebra \(R_A\), introduced by Auslander, and further studied by Smalø. The algebra \(R_A\) contains \(A\) as an idempotent subalgebra and is quasihereditary with respect to a heredity chain constructed by Dlab and Ringel. In this talk we will discuss the nice properties of \(R_A\) that stem from this heredity chain.

## January 28th 2015, 13:15 - 14:15 in 734

**Title:** An intersection-dimension formula from decorated marked surfaces

**Speaker:** Yu Zhou

**Abstract:** This is a joint work with Yu Qiu. For a triangulated decorated marked surface S, there is an associated differential graded algebra whose finite dimensional derived category D is a 3-Calabi-Yau triangulated category. Under a bijection between closed arcs in S and spherical objects in D, we give a formula connecting intersection numbers between closed arcs and dimensions of graded morphism spaces between the corresponding objects.

## January 21st 2015, 13:15 - 14:15 in 734

**Title:** Periodicity and weighted surface algebras

**Speaker:** Karin Erdmann

**Abstract:** This will discuss some results on selfinjective algebras for which all simple modules are \(Omega\)-periodic. In particular we describe a new class of algebras constructed from triangulations of surfaces. (joint with A. Skowronski)

## January 14th 2015, 13:15 - 14:15 in 734

**Title:** Recent developments: n-angulated categories

**Speaker:** Petter Andreas Bergh

**Abstract:** In a paper appearing last year, Geiss, Keller and Oppermann introduced higher analogues of triangulated categories, called n-angulated categories. They appear for instance when one studies certain cluster tilting subcategories of triangulated categories. In this talk, we give an overview of some of the recent developments. This is joint work with Marius Thaule.