Köln Algebra and Representation Theory Seminar

Abstract: (joint work with Claude Ciblis, Marcelo Lanzilotta and Eduardo Marcos) In this talk I will introduce the τ-tilting Hochschild cohomology in degree one of a finite dimensional k-algebra A, where k is a field. The excess of A is the difference between the dimensions of the τ-tilting Hochschild cohomology in degree one and the dimension of the usual Hochschild cohomology in degree one. One of the main results is that for bound quiver algebra A=KQ/I such that its excess is zero, the Hochschild cohomology in degree two HH^2(A) is isomorphic to the space Hom_{KQ-KQ}(I/I^2, A). This may be useful to determine when HH^2(A)=0 for these algebras. We compute the excess for hereditary, radical square zero and monomial triangular algebras. For bound quiver algebra A, a formula for the excess is obtained. We also give a criterion for A to be τ-rigid.

Abstract: In joint work with Esther Banaian, Raphael Bennett-Tennenhaus, and Karin Jacobsen, we define a geometric model for certain generalizations of gentle algebras. We consider semilinear locally gentle algebras - (potentially infinite-dimensional) gentle algebras where we allow division rings on vertices and we associate automorphisms of these division rings to each arrow with certain multiplication rules. In this talk, we will demonstrate how one can utilize 🍩 topological tools 🍩 to visualize objects in the module category for such algebras. Classically, it has been shown that one can model various quiver algebras and associated categories using surfaces. In the simplest example, the category of indecomposable type A quiver representations can be viewed using triangulations of polygons. For our case, we model our algebras with punctured surfaces that have been cut up and stitched back together at the seams. This process of cutting and stitching the surface is a geometric realization of an algebraic result of Zembyk and it allows us to endow an existing geometric model for locally gentle algebras with a semilinear twist. 

Abstract: Let T be a triangulated category and R a rigid subcategory of T. Iyama--Yang provide a mild technical condition that lets us explicitly compute the Verdier quotient T/thick(R). Although localisation theory of extriangulated categories is more complicated, there are good reasons to generalise Iyama--Yang's work in that direction, for example in order to redefine tau-cluster morphism categories using (ex)triangulated machinery. In this talk, we describe such a generalisation, formulating a technical condition in terms of generalised Hovey twin cotorsion pairs. For 0-Auslander extriangulated categories (for example a category of projective presentations), it suffices that the rigid subcategory R admits Bongartz completions. We will also mention the connection with tau-tilting reduction and tau-cluster morphism categories.

Abstract: In this talk, I will show over finite field, the gentle one-cycle algebra Λ(n-1,1,1) has Hall polynomials. The Hall polynomials are explicitly given for all triples of indecomposable modules, and as a consequence, the Ringel--Hall Lie algebra of Λ(n-1,1,1) is shown to be isomorphic to a Lie algebra of type BC^+. This is a joint work with Dong Yang. 

Abstract: I will describe eigenvalues and certain eigenobjects (a.k.a. Calabi-Yau objects) for the Serre endofunctor of the bounded derived category of  the principal block of the BGG category O associated to a triangular decomposition of a semi-simple complex Lie algebra. Essential ingredients of the story are the action of the bicategory of projective functors (a.k.a. the Hecke category) and the induced action of the 2-braid group as well as the Auslander regularity of category O.

Abstract: We give a geometric model for any length heart in the derived category of a gentle algebra, which is equivalent to the module category of some gentle algebra. To do this, we deform the geometric model for the module category of a gentle algebra given in [BC21], and then embed it into the geometric model of the derived category given in [OPS18], in the sense that each so-called zigzag curve on the surface represents an indecomposable module as well as the minimal projective resolution of this module. A key point of this embedding is to give a geometric explanation of the dual between the simple modules and the projective modules.

Such a blend of two geometric models provides us with a handy way to describe the homological properties of a module within the framework of the derived category. In particular, we realize any higher Yoneda-extension as a polygon on the surface, and realize the Yoneda-product as gluing of these polygons.

Abstract: Brauer graph algebras are finite dimensional algebras that are constructed from a combinatorial data called a Brauer graph. Kauer proved that one can obtain derived equivalences of Brauer graph algebras from a particular move of one edge in the corresponding Brauer graph. Moreover, this derived equivalence is described by a tilting object which is in fact a silting mutation. In this talk, I will be interested in skew Brauer graph algebras which generalize the class of Brauer graph algebras. These algebras are constructed from a Brauer graph with possible "degenerate" edges. I will explain how Kauer's results can be generalized to the move of multiple edges and can also be generalized for skew Brauer graph algebras.

Abstract: For a finite-dimensional algebra A, being Auslander-Gorenstein is a homological condition which implies many interesting properties for the algebra and for certain subcategories of mod(A). In this talk, we will consider three well-known classes of algebras; namely, gentle, Nakayama, and monomial algebras, and aim to understand what the Auslander-Gorenstein property means in these settings. First, we will try to find a combinatorial characterisation of this homological condition, which leads us to a new class of examples of Auslander-Gorenstein algebras. Second, I will present a surprising new homological characterisation of the Auslander-Gorenstein property for these algebras. For this, a bijection between indecomposable projective and injective A-modules introduced by Auslander and Reiten plays a central role.

Abstract: We explain recent work with Omar Kidwai where we compute the Donaldson--Thomas invariants for a certain quiver with potential dubbed the ``barbell quiver with potential''.

The motivation is to compute the Donaldson--Thomas invariants arising in the correspondence between quadratic differentials and stability conditions first shown by Bridgeland and Smith, and later refined by Christ, Haiden, and Qiu.