Representation Theory Group Seminar - Cologne

Summer Semester 2024

08.04 Michael Tsironis (VU Amsterdam, online)

Title: Skein relations for punctured surfaces

Abstract: Given a marked bordered surface S and a triangulation T, one can associate a Cluster algebra and a Jacobian algebra. In the classical setting where all the marked points lie on the boundary, snake graph calculus is a very helpful combinatorial tool, which was used by İlke Çanakçı and Ralf Schiffler (2015) to give an alternative proof of the so-called skein relations, which are some identities in terms of cluster variables. Recently, Jon Wilson (2020) constructed a new type of snake graph, the so-called loop graph, which allows one to easily work in the setting, where we are also allowing marked points (punctures) in the interior of the surface S. In this talk, we will describe how one can associate a module in the Jacobian algebra, to every given loop graph, and how we can use this construction to prove skein relations in this broader setting.

22.04 Xiuli Bian (ECNU and Cologne)

Title: Classifying recollements of derived module categories for derived discrete algebras

Abstract: We study a class of derived discrete Nakayama algebras. All indecomposable compact objects in the derived module category are determined and all recollements generated by the indecomposable compact exceptional objects are classified. It reveals that all such recollements are derived equivalent to stratifying recollements. As a byproduct, this confirms a question due to Xi for these recollements.

29.04 Calvin Pfeifer (Cologne)

Title: Torsion-free and divisible modules from schemes of modules.

Abstract: Motivated by the theory of large modules over tame hereditary algebras, developed by Ringel and Reiten-Ringel, we will present a generalisation of torsion-free and divisible modules to arbitrary finite-dimensional algebras with respect to a fixed weight from the dual Grothendieck group. Crawley-Boevey noted that points of schemes of modules give rise to endofinite modules (i.e. modules of finite-length over their endomorphism algebra), and we will sketch an upper-semicontinuity argument to construct torsion-free and divisible modules from families of finite-dimensional stable modules. We will illustrate the construction by a gentle algebra associated to a Cartan matrix of extended type C_2.