# Seminars in Algebra

## Fall 2016

## December 7th 2016, 14:15 in 734

**Title:** Spherical objects and the higher cluster category of the cyclic apeirotope

**Speaker:** Julian Külshammer

**Abstract:** This is a report on ongoing joint work with Gustavo Jasso. Let D be the triangulated category associated to a spherical object of dimension m greater than or equal to 2. By work of Jørgensen, this is an m-Calabi-Yau triangulated category with almost split triangles classically generated by a spherical object. Moreover, its Auslander-Reiten quiver has m-1 connected components of type ZA-infinity. Building upon work of Amiot, Guo, Keller, and Oppermann-Thomas, for each positive integer d we construct an md-Calabi-Yau weakly (d+2)-angulated category with almost split (d+2)-angles. Moreover, its higher Auslander-Reiten quiver has m-1 connected components of higher mesh type A-infty. For m=2, our construction is analogous to the cluster category of type A-infinity introduced by Holm-Jørgensen.

## November 17th 2016, 14:15 in 656

**Title:** Higher preprojective algebras and superpotentials

**Speaker:** Louis-Philippe Thibault

**Abstract:** In the framework of Iyama’s program on higher Auslander-Reiten theory, the notion of preprojective algebra was extended to algebras of higher global dimension. In this talk, we give a quiver construction of higher preprojective algebras over basic Koszul n-representation-infinite algebras. We then explain that such algebras can be presented using a higher superpotential, which encodes nicely their relations. We also provide background notions and give examples.

## September 8th 2016, 14:30 in 656

**Title:** Thick subcategories of d-abelian categories (report on joint work with Martin Herschend and Laertis Vaso)

**Speaker:** Peter Jørgensen

**Abstract:** Let d be a positive integer. The notion of d-abelian categories was introduced by Jasso. Such a category does not have kernels and cokernels, but rather d-kernels and d-cokernels which are longer complexes with weaker universal properties. Canonical examples of d-abelian categories are d-cluster tilting subcategories of abelian categories. We introduce the notion of thick subcategories of d-abelian categories and show a classification of the thick subcategories of a family of d-abelian categories associated to quivers of type A_n. If time permits, we show how thick subcategories are in bijective correspondence with a particularly nice class of objects which we call thick generators.

## September 8th 2016, 13:15 in 656

**Title:** Special Algebras

**Speaker:** Sibylle Schroll

**Abstract:** Special algebras comprise the well-studied (tame) special biserial algebras and their (wild) generalisations, the special multiserial algebras. Well-known special biserial algebras include Brauer graph algebras which arise out of the modular representation theory of finite groups as well as string algebras, such as Nakayama or gentle algebras. In this talk, we will give a brief overview of many of the properties of special biserial algebras motivating similar results for special multiserial algebras.

## Spring 2016

## June 15th 2016, 13:15 - 14:15 in 734

**Title:** A representation theoretic approach to exceptional sheaves on the projective n-space.

**Speaker:** Dan Zacharia

**Abstract:** See the abstract.

## June 15th 2016, 14:30 - 15:30 in 734

**Title:** Mesh categories of type A_infty and tubes in higher Auslander-Reiten theory

**Speaker:** Gustavo Jasso

**Abstract:** This is a report on ongoing work with Julian Külshammer. We construction higher analogues of mesh categories of type A_infty and of the tubes from the viewpoint of Iyama's higher Auslander-Reiten theory. Our construction relies on unpublished work by Darpö and Iyama. We sketch a conjectural construction which relates our categories to spherical objects and to cluster tubes.

## June 9th 2016, 10:00 - 11:00 in 734

**Title:** Higher representation theory: 2-representations of finitary 2-categories

**Speaker:** Volodymyr Mazorchuk

**Abstract:** This series of lectures is aimed as a general introduction to the theory of finitary additive 2-representations of finitary 2-categories. The main emphasize will be made on classification problems arizing in this theory, primarily for simple transitive 2-representations. The idea is to illustrate the whole theory on the basis of two small (but non-trivial) examples.

## June 8th 2016, 10:00 - 11:00 in 656

**Title:** Higher representation theory: 2-representations of finitary 2-categories

**Speaker:** Volodymyr Mazorchuk

**Abstract:** This series of lectures is aimed as a general introduction to the theory of finitary additive 2-representations of finitary 2-categories. The main emphasize will be made on classification problems arizing in this theory, primarily for simple transitive 2-representations. The idea is to illustrate the whole theory on the basis of two small (but non-trivial) examples.

## June 7th 2016, 10:00 - 11:00 in 656

**Title:** Higher representation theory: 2-representations of finitary 2-categories

**Speaker:** Volodymyr Mazorchuk

**Abstract:** This series of lectures is aimed as a general introduction to the theory of finitary additive 2-representations of finitary 2-categories. The main emphasize will be made on classification problems arizing in this theory, primarily for simple transitive 2-representations. The idea is to illustrate the whole theory on the basis of two small (but non-trivial) examples.

## June 6th 2016, 10:00 - 11:00 in 656

**Title:** Higher representation theory: 2-representations of finitary 2-categories

**Speaker:** Volodymyr Mazorchuk

**Abstract:** This series of lectures is aimed as a general introduction to the theory of finitary additive 2-representations of finitary 2-categories. The main emphasize will be made on classification problems arizing in this theory, primarily for simple transitive 2-representations. The idea is to illustrate the whole theory on the basis of two small (but non-trivial) examples.

## June 1st 2016, 14:15 - 15:15 in 734

**Title:** Lattice structure of preprojective algebras and Weyl groups

**Speaker:** Osamu Iyama

**Abstract:** Tilting theory of the preprojective algebra A of an acyclic quiver Q categorifies the corresponding Coxeter group W. When Q is Dynkin, there exists an isomorphism of lattices between W with the opposite weak order and torsion classes of A. This gives bijections between join-irreducible elements in W and bricks of A. If time permits, for type A, we characterize the lattice quotients of W coming from algebra quotients of A. This is a joint work with N. Reading, I. Reiten and H. Thomas.

## April 20th 2016, 13:00 - 14:00 in 656

**Title:** A generalized Dade's Lemma for local rings

**Speaker:** David Jorgensen

**Abstract:** In this talk we discuss a generalized Dade's Lemma for quotients of local rings by ideals generated by regular sequences. That is, given a pair of finitely generated modules over such a ring with algebraically closed residue field, we prove a sufficient (and necessary) condition for the vanishing of all higher Ext or Tor of the modules. This condition involves the vanishing of all higher Ext or Tor of the modules over all quotients by a minimal generator of the ideal generated by the regular sequence. This is based on joint work with Petter Bergh.

## April 13th 2016, 13:15 - 14:15 in 734

**Title:** Frames and triangulated categories

**Speaker:** William Sanders

**Abstract:** See abstract.

## April 6th 2016, 13:15 - 14:15 in 734

**Title:** Frieze patterns and cluster theory

**Speaker:** Karin Baur

**Abstract:** Cluster categories and cluster algebras can be described via triangulations of surfaces. In type A, such triangulations lead to frieze patterns in the sense of Conway and Coxeter. We illustrate this connection and study properties of frieze patterns.

## March 15th 2016, 13:10 - 14:10 in 656

**Title:** Hochschild cohomology of relation extension algebras

**Speaker:** Rachel Taillefer

**Abstract:** See the abstract.

## February 24th 2016, 14:25 - 15:25 in 734

**Title:** Is functional analysis a special case of tilting theory?

**Speaker:** Sven-Ake Wegner

**Abstract:** See the abstract

## February 24th 2016, 13:15 - 14:15 in 734

**Title:** Analogues of Nakayama algebras in higher Auslander-Reiten theory

**Speaker:** Julian Külshammer

**Abstract:** In higher dimensional Auslander-Reiten theory one studies algebras whose module categories have subcategories admitting sequences of length d+2 which are analogous to classical almost split sequences. As in classical Auslander-Reiten theory, the ones of smallest possible global dimension (which in this case is d) and selfinjective cases have been studied first. The ones of global dimension d are called d-representation-finite d-hereditary algebras. In this talk we present joint work with Gustavo Jasso which allows to construct higher analogues of all Nakayama algebras starting from higher analogues of the hereditary and selfinjective Nakayama algebras, which were constructed by Iyama and Darpö-Iyama. We also indicate which properties they share and at which points they differ from classical Nakayama algebras.

## February 17th 2016, 13:15 - 14:15 in 734

**Title:** Stable categories of preprojective algebras and c-sortable elements

**Speaker:** Yuta Kimura

**Abstract:** Let Q be a finite acyclic quiver and Pi the preprojective algebra of Q. Buan-Iyama-Reiten-Scott introduced the factor algebra Pi_w associated with a element w in the Coxeter group of Q. They show that the stable category of the Cohen-Macaulay Pi_w modules has cluster tilting objects. In this talk, we regard Pi_w as a graded algebra and show that the stable category of the graded Cohen-Macaulay Pi_w modules has tilting objects if w is a c-sortable element.

## February 3rd 2016, 13:15 - 14:15 in 734

**Title:** Endomorphism algebras of 2-term silting complexes

**Speaker:** Yu Zhou

**Abstract:** Classical tilting theory gives a way to relate the module categories of two algebras A and B, where B is the endomorphism algebra of a tilting A-module T. For instance, there is an induced cotilting B-module C whose endomorphism algebra is isomorphic to A; the torsion class (resp. the torsion free class) induced by T is equivalent to the torsion free class (resp. the torsion class) induced by C; the Auslander-Reiten theory of mod B can be described in terms of the Auslander-Reiten theory of mod A. Hence one can study mod B via mod A. In this talk, I will show how to generalize these results to the 2-term silting complex setting. This is a joint work with Aslak Bakke Buan.

## January 27th 2016, 13:15 - 14:15 in 734

**Title:** Survey on 2-representation finite algebras

**Speaker:** Martin Herschend

**Abstract:** In higher dimensional Auslander-Reiten theory one studies algebras whose module categories have subcategories with suitable homological properties. The first case to consider are so called d-representation finite algebras. These are algebras of global dimension d having d-cluster tilting subcategories with finitely many indecomposables. Thus 1-representation finite algebras are hereditary representation finite algebras, which by Gabriel's Theorem, are given by Dynkin quivers. For 2-representation finite algebras no similar classification is known. Nevertheless, every 2-representation finite algebra is given by certain datum called a selfinjective quiver with potential and cut. I will give a survey the known sources of 2-representation finite algebras and show how the corresponding selfinjective quivers with potential can be described.

## January 20th 2016, 13:15 - 14:15 in 734

**Title:** Matrix factorizations and complete intersections

**Speaker:** Petter Andreas Bergh

**Abstract:** By a classical result of Eisenbud, every minimal free resolution over a hypersurface originates from a matrix factorization over the ambient regular ring. Hypersurfaces are precisely the complete intersections of codimension one, and so it is natural to look for connections between matrix factorizations and more general complete intersections. In this talk, we shall see that one can realize certain triangulated subcategories of the singularity category of a complete intersection as homotopy categories of matrix factorizations. This is based on joint work with David Jorgensen.