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. 

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