# Seminars in Algebra

## Fall 2014

## December 10th 2014, 13:15 - 14:15 in 734

**Title:** Notions of representation stability II

**Speaker:** Peter Patzt

**Abstract:** See 'Notions of representation stability I'.

## December 3rd 2014, 13:10 - 14:00 in 734

**Title:** Notions of representation stability I

**Speaker:** Peter Patzt

**Abstract:** In two parts I want to give an overview of different notions of representation stability. The goals of my talks will be to describe and compare these notions, focusing on multiplicity stability defined by Church and Farb in [CF] and central stability defined by Putman in [P]. I will also give some background on how these notions arose and their connection to homological stability as well as other applications. In the end I want to give some technical details and if time permits a personal view on the theory. [CF] Church, Farb: 'Representation theory and homological stability', arXiv:1008.1368; [P] Putman: 'Stability in the homology of congruence subgroups', arXiv:1201.4876.

## October 15th 2014, 13:10 - 14:10 in 734

**Title:** Derived invariants for surface algebras

**Speaker:** Claire Amiot

**Abstract:** Let (S,M) be unpunctured surface with marked points, with genus g and b boundary components. A surface algebra arises from a cut of an ideal triangulation of (S,M). In a joint work with Yvonne Grimeland, we associate to any surface algebra and any generating set of the fundamental group of S, an element in mathbb{Z}^{2g+b}. We show that this element determines the derived equivalence class of the algebra up to homeomorphism of the surface. In this talk I will explain this result, the main ingredients of the proof, and the information we can deduce on the corresponding derived categories.

## October 8th 2014, 13:10 - 14:10 in 734

**Title:** Braid groups and quiver mutation

**Speaker:** Robert Marsh

**Abstract:** This is joint work with Joseph Grant (Leeds). Associated to each Dynkin diagram is a corresponding Artin braid group; in type A_n this group is the usual braid group on n+1 strands. In the simply-laced case, we associate a presentation for the Artin braid group to each quiver arising in the corresponding cluster algebra without coefficients. The presentations are compatible with quiver mutation. We give a topological interpretation of the new generators and a categorical interpretation of the presentations, with the generators acting as spherical twists at simple modules on derived categories of Ginzburg dg-algebras.

## September 23rd 2014, 15:15 - 16:15 in 734

**Title:** Silting modules

**Speaker:** Lidia Angeleri-Hügel

**Abstract:** The talk is devoted to a class of modules, called silting, which yield a common generalization of support τ-tilting modules over a finite dimensional algebra and of (not necessarily finitely generated) tilting modules over an arbitrary ring. We will see that some well known results on existence of approximations and complements can be extended to this new concept. Moreover, silting modules are in bijection with 2-term silting complexes and with certain t-structures and co-t-structures in the derived module category. Finally, we will show that there is a close relationship between silting modules and ring epimorphisms. This is a report on joint work with Frederik Marks and Jorge Vitória.

## September 17th 2014, 13:10 - 14:10 in 734

**Title:** Hochschild cohomology from a topologist's perspective

**Speaker:** Markus Szymik

**Abstract:** I will first present a class of topological problems (and their solution) that are related to Hochschild cohomology by the usual formula. I would then like to discuss how to tighten this connection by using a dictionary that translates between topology and algebra. I intend to end with a list of ensuing problems.

## September 10th 2014, 14:45 - 15:45 in 1329

**Title:** A discrete introduction

**Speaker:** David Pauksztello

**Abstract:** In this talk, we will provide a survey of the structure of discrete derived categories. The notion of a discrete derived category was introduced by Vossieck, who classified the algebras with discrete derived categories, the so-called derived-discrete algebras. Bobinski, Geiss and Skowronski then determined a canonical form for the derived equivalence class of a derived-discrete algebra and described the structure of their Auslander-Reiten quivers explicitly. Their structure is simple enough to enable concrete calculations, but sufficiently non-trivial to manifest interesting phenomena. This makes them an ideal natural laboratory to study derived representation theory. In this talk, I will further elaborate on the structure of these categories, explaining how to classify bounded t-structures, what kinds of objects can occur, determining the auto-equivalence group, and sketching further nice properties. This is a report on joint work with Nathan Broomhead and David Ploog. The pun in the title is due to Robert Marsh.

## September 10th 2014, 13:30 - 14:30 in 1329

**Title:** Hom-configurations in negative Calabi-Yau triangulated categories

**Speaker:** Raquel Coelho Simoes

**Abstract:** Calabi-Yau triangulated categories appear in many branches of mathematics and physics, for example as cluster categories in representation theory. Much work has been done on understanding triangulated categories of positive CY dimension, particularly those which are 2-CY or 3-CY. Thus far, little is understood about triangulated categories of negative CY dimension. Examples of such categories arise out of the work of Riedtmann on the classification of selfinjective algebras and were one of the original motivations in the development of cluster-tilting theory. In this setting Hom-configurations are the natural objects of study, and their behaviour in a certain orbit category C of the derived category with negative CY dimension is highly reminiscent of that of cluster-tilting objects. In this talk, we consider a generalization of Hom-configurations in triangulated categories generated by spherical objects with negative CY dimension. We will give a combinatorial classification of these configurations and explain links with noncrossing partitions. Along the way, we obtain a geometric model for the higher versions of the orbit category C.

## September 2nd 2014, 14:45 - 15:45 in 1329

**Title:** Bismash products as category algebras

**Speaker:** Karin Erdmann

**Abstract:** This is a class of Hopf algebras with interesting tensor products, amongst these are the algebras studied by Benson and Witherspoon. (Joint work with S. Danz)

## September 2nd 2014, 13:30 - 14:30 in 1329

**Title:** The Nakayama automorphism for self-injective preprojective algebras

**Speaker:** Joseph Grant

**Abstract:** The preprojective algebra of a graph is finite-dimensional if and only if the graph is Dynkin, and it is known in this case that the preprojective algebra is self-injective. I will give a new proof of the self-injectivity of this algebra, using the Baer-Geigle-Lenzing description of the preprojective algebra, and will describe the Nakayama automorphism. This description generalizes to higher preprojective algebras, as studied by Iyama and collaborators.

## August 28th 2014, 13:15 - 14:15 in 922

**Title:** Cluster exchange groupoids and braid group action of Dynkin type

**Speaker:** Yu Qiu

**Abstract:** We introduce m-cluster exchange groupoid via m-cluster categories for a Dynkin quiver Q. We show that its point group is the braid group Br(Q) associated to Q and its universal cover is the exchange graph EG of hearts in the corresponding (m+1)-Calabi-Yau category D associated to Q. As applications, we show that there is a faithful braid group action Br(Q) on D and EG admits a Garside groupoid structure. (joint work with Jon Woolf)

## August 20th 2014, 13:15 - 14:15 in 734

**Title:** Grassmannian cluster categories

**Speaker:** Alastair D King

## Spring 2014

## June 18th 2014, 15:30 - 16:30 in 734

**Title:** Torsion pairs in a triangulated category generated by a spherical object

**Speaker:** David Pauksztello

**Abstract:** This is a report on joint work with Raquel Coelho Simoes (Lisbon). Recently, Holm, Jorgensen and Yang studied triangulated categories T_w generated by a w-spherical object. The category T_2 can be thought of as a cluster category of type A infinity and extensions in this category can be computed by crossings of arcs in an appropriate geometric/combinatorial model. Using this model, Ng classified torsion pairs in T_2 in terms of Ptolemy diagrams. We extend extend this classification to arbitrary w. When w>0, we obtain a classification of torsion pairs in terms of Ptolemy diagrams. When w<0, the combinatorics are more complicated and we obtain a classification in terms of modified Ptolemy diagrams.

## June 18th 2014, 14:15 - 15:15 in 734

**Title:** From triangulated categories to module categories via model categories

**Speaker:** Yann Palu

**Abstract:** Motivated by the theory of cluster algebras, Buan-Marsh-Reiten proved that some quotients of cluster categories are module categories. More generally, some subquotients of some Hom-finite triangulated categories are module categories. In a recent paper, A. Buan and R. Marsh proved that these module categories can also be recovered as certain localisations of the triangulated categories under consideration. Our aim is to give a homotopical algebra point of view on their result.

## June 4th 2014, 14:15 - 15:15 in 734

**Title:** Aisles in derived categories of finite type hereditary algebras

**Speaker:** Hugh Thomas

**Abstract:** Let Q be a Dynkin quiver. Colin Ingalls and I showed that torsion classes in kQ-mod correspond to the inversion sets of c-sortable elements (in the sense of Reading) of the Weyl group associated to Q. I will report on joint work with Christian Stump and Nathan Williams, in which we consider the extension of this problem to aisles in the bounded derived category of H-mod, for H a hereditary finite representation type algebra. We introduce a notion of c-sortability for Artin groups, and we show that the generating, separated aisles correspond to inversion sets of c-sortable elements in the Artin group corresponding to W.

## May 28th 2014, 14:15 - 15:15 in 734

**Title:** Approximations of totally acyclic complexes

**Speaker:** David Jorgensen

**Abstract:** Let Q be a commutative local Gorenstein ring, I an ideal of Q having finite projective dimension, and R = Q/I. In this talk we define an adjoint pair of exact functors between the homotopy category of totally acyclic complexes over Q and that over R. As a consequence, one obtains a precise notion of approximation of a totally acyclic complex over R by a totally acyclic complex over Q. In particular, we show that one may approximate an arbitrary totally acyclic complex over a complete intersection by matrix factorizations. This is based on joint work with Kristen Beck, Petter Bergh, and Frank Moore.

## May 8th 2014, 14:15 - 15:15 in 734

**Title:** Spaces of stability conditions for Calabi-Yau A_2 quivers

**Speaker:** Yu Qiu

**Abstract:** (Joint work with Tom Bridgeland and Tom Sutherland) We compute the spaces of stability conditions of the derived category of an A_2 quiver and its Calabi-Yau(-N) variations. Firstly we obtain uniform results for all N: the space of stability conditions quotiented by the action of the spherical twists is independent of N, although the identiﬁcation maps are highly non-trivial. Secondly, there is a close link between our spaces of stability conditions and the polynomial Frobenius manifold corresponding to the A_2 root system (or rather, its almost dual): in fact the Frobenius structure encodes and is in some sense equivalent to the identiﬁcations between our stability spaces for various N. Finally, we see that the usual (A_2) quiver should be considered as Calabi-Yau-infinity.

## April 30th 2014, 14:15 - 15:15 in 734

**Title:** Derived categories of cluster-tilted algebras of type A

**Speaker:** Fedra Babaei

**Abstract:** Let Q be a quiver which can be obtained from a Dynkin quiver of type A via quiver mutations as defined by Fomin and Zelevinsky. For a field k, let Lambda=kQ/(R) be a cluster-tilted algebra of type A. Any indecomposable object in K^b(Proj Lambda) corresponds to a homotopy string or a homotopy band in Q. We classify the AR-components of K^b(Proj Lambda) via homotopy strings and homotopy bands. At last, we consider a class of homotopy string called direct homotopy string and we give an explicit description of all self-orthogonal indecomposable objects corresponding to direct homotopy strings in Q. In addition, we describe all self-orthogonal indecomposable objects in the AR-components of type ZA_infty.

## April 2nd 2014, 14:15 - 15:15 in 734

**Title:** Generic cluster characters

**Speaker:** Pierre-Guy Plamondon

**Abstract:** Over a finite-dimensional algebra, every module has a projective presentation. Conversely, given two finite-dimensional projective modules, one can define the notion of generic morphism between them; the generic cokernel thus obtained has nice properties that were used Geiss, Leclerc and Schröer in the categorification of cluster algebras. In this talk, we will present some properties of these generic notions in the setting of 2-Calabi-Yau triangulated categories, with one eye kept on applications to cluster algebras.

## March 26th 2014, 14:15 - 15:15 in 734

**Title:** Betti polynomials

**Speaker:** Petter Andreas Bergh

**Abstract:** Given a finite dimensional algebra and two modules, how do the dimensions of the cohomology groups behave? We shall see that when certain cohomological finiteness conditions hold, then the dimensions are eventually given by polynomials.

## March 12th 2014, 14:15 - 15:15 in 734

**Title:** Enriched Brown representability

**Speaker:** Johan Steen

**Abstract:** Brown-type representability theorems give us criteria for when certain functors are of the form Hom(-,X). Several versions for various source and target categories exist, both topological and algebraic in nature. Oftentimes the Hom-sets carry extra structure, so it is natural to ask whether one can make sense of an enriched representability theorem. For motivation, I will discuss some newer variants of representability theorems (for triangulated categories), before introducing enriched categories and explaining the contents of our main result. This is joint work with Greg Stevenson.

## February 5th 2014, 14:15 - 15:15 in 734

**Title:** A short proof for the Morita invariance of the Gerstenhaber bracket in Hochschild cohomology

**Speaker:** Reiner Hermann

**Abstract:** In 2004, Bernhard Keller proved that the Gerstenhaber bracket in Hochschild cohomology is a derived invariant. By utilizing Jeremy Rickard’s characterization of derived equivalences in terms of invertible bimodule complexes, and James Stasheff’s interpretation of Hochschild cohomology as coderivations, Keller identified the Lie algebra which underlies the Hochschild cohomology ring with the Lie algebra associated to the derived Picard group. In this talk, we will explain how to use a generalization of Stefan Schwede’s exact sequence interpretation of the Gerstenhaber bracket to obtain a short (and completely different) proof of the invariance in the more specific case of Morita equivalent algebras.

## January 29th 2014, 14:15 - 15:15 in 734

**Title:** The Homotopy Theory of (∞,1)-Categories

**Speaker:** Magnus Hellstrøm-Finnsen

**Abstract:** The study of stable (∞,1)-categories can for example be motivated from that many prominent examples of triangulated categories are given almost by definition as homotopy categories of stable (∞,1)-categories. The objective for this speech is however to discuss some of the ideas and notions in the theory of (∞,1)-categories in order to state and understand the fact that the homotopy category of a stable (∞,1)-category is a triangulated category. Here, the study of (∞,1)-categories is formalised by quasi-categories, having the benefit that many notions for our purpose adapt intuitively and well-motivated from the theory of ordinary categories. Quasi-categories are defined to be simplicial sets with the inner horn filler property. This speech is based on my master project, supervised by Petter Andreas Bergh and Marius Thaule.

## January 22nd 2014, 14:15 - 15:15 in 734

**Title:** The classification of τ-tilting modules over Nakayama algebras

**Speaker:** Takahide Adachi

**Abstract:** In this talk, we study τ-tilting modules over Nakayama algebras. We establish bijections between τ-tilting modules, triangulations of a polygon with a puncture and certain integer sequences. These bijections are an analog of a result of Buan-Krause.