Köln Algebra and Representation Theory Seminar

Abstract: 

Abstract: In this talk, we propose a stable and a τ-reduced version of the second Brauer-Thrall conjecture. The former is a slight strengthening of a brick version of the second Brauer-Thrall conjecture introduced by Mousavand and Schroll-Treffinger-Valdivieso. The latter is stated in terms of Geiß-Leclerc-Schröer's generically τ-reduced components and provides a geometric interpretation of a question raised by Demonet. We outline implications among these conjectures and relate them to recent variations of tameness in stability and τ-tilting theory. It follows from Schroll-Treffinger-Valdivieso’s work that the conjectures are true for special biserial algebras, and we confirm them for Geiß-Leclerc-Schröer’s (GLS) algebras associated to valued quivers. If time permits, we demonstrate that the in general representation wild GLS algebras of affine type are still „tame“ from a τ-tilting perspective. 

Abstract: Adachi, Iyama and Reiten developed τ-tilting theory to mirror the properties of mutation seen in cluster algebras. The theory gives a generalisation of classical tilting modules using the Auslander-Reiten translation τ, and one studies distinguished pairs of objects in the module category of a finite-dimensional algebra known as τ-rigid pairs. An important result from the theory is the correspondence between functorially finite torsion classes, maximal τ-rigid pairs and 2-term silting complexes (amongst others).

Meanwhile, higher homological algebra has since its introduction by Iyama become a very active field of research, and many authors have generalised notions to the higher homological setting, including both torsion classes and τ-rigid pairs. This talk is a report on work in progress investigating the relationship between higher torsion classes, silting objects and maximal τ_d-rigid pairs, describing explicit correspondences.The talk is based on joint work with August, Haugland, Kvamme, Palu and Treffinger

Abstract: The string coproduct on the homology of the free loop space of a manifold is one of the major operations in string topology. We will construct an algebraic analogue of the string coproduct. The algebraic framework is built upon the Hochschild chain complex of a smooth dg category equipped with a pre-Calabi-Yau structure and a trivialisation of the Chern character of the diagonal bimodule.

We show that under a mild condition our algebraic coproduct is graded cocommutative and coassociative. This is a joint work with Manuel Rivera and Alex Takeda. 

Abstract: It is well known that the deformations of algebras are in correspondence with the Maurer-Cartan elements. In this talk, for a gentle algebra A, we will consider the $L_\infty$ structure of the Bardzell complex and establish conditions on the quiver of A that allow us to ensure the nilpotency of the brackets and calculate the Maurer-Cartan elements of A.

This talk is based on a joint work with Monique Müller, María Julia Redondo, and Pamela Suarez. 

Abstract: In this talk, we will review a certain symmetry of silting quivers introduced by Aihara and the speaker in 2022, and the sign decomposition, which was introduced by Aoki in 2018 to classify torsion classes of radical square zero algebras and extended by Aoki, Higashitani, Iyama, Kase, Mizuno in 2022 to study fans and polytopes in $\tau$-tilting theory. As an application, we will provide a classification of 2-term silting finite Borel-Schur algebras.

Abstract: I will define a family of G-spaces associated to a group G with a fixed generators which generalize Coxeter complexes and study the representations arising in their cohomology for some special cases.

This is joint with Victor Reiner.

Abstract: Let g be a finite dimensional simple Lie algebra over C with the triangular decomposition g = n^- + h + n. String cones were introduced by Littelmann and Berenstein-Zelevinsky and the integer points of string cones parametrize dual canonical bases in quantum coordinate rings A_q(n). These integer points coincide with polyhedral realizations for Kashiwara crystals of the negative part U_q(n^-) of quantum groups. It is a natural problem to find an explicit form of the string cone. In the context of geometric crystals, a regular function \Phi on an algebraic group called Berenstein-Kazhdan decoration function was introduced. From the explicit form of \Phi, one can immediately obtain an explicit form of the string cone. 

In this talk, we briefly review the representation theory of finite dimensional simple Lie algebras, quantum groups, string cones and geometric crystals. After that, when g is of classical type, we will give an algorithm to explicitly compute \Phi. This is a joint work with Gleb Koshevoy and Toshiki Nakashima.

Abstract: In a quiver-representation-theoretic approach to cluster algebras, Derksen-Weyman-Zelevinsky defined mutations of quivers with potential and their representations around 16 years ago. Among other things, they showed that their mutations send \tau-rigid pairs to \tau-rigid pairs, and are compatible with Adachi-Iyama-Reiten mutations (which appeared a few years later).

On the other hand, in an attempt to complete the set of cluster monomial to a basis of the cluster algebra, Geiss-Leclerc-Schröer introduced the notion of \tau-reduced irreducible component as a generalization of \tau-rigidity. In this talk I will sketch joint work with Christof Geiss and Jan Schröer, in which we show that DWZ's mutations send \tau-reduced components to \tau-reduced components (thus generalizing Derksen-Weyman-Zelevinsky's result) and realize DWZ's mutations as regular maps on dense subsets of these components. 

Abstract: Classic Auslander-Reiten theory is a neat tool used to paint a portrait of the category of modules over an Artinian ring. Nakayama functors play an important role in this painting. In suitable settings, the theory generalises to abelian categories, triangulated categories and their subcategories. In this talk, we will construct Nakayama functors on proper abelian subcategories. These categories, defined by Jørgensen in 2022, are generalisations of hearts of t-structures.

Talk is based on the following arXiv preprint: arXiv:2312.07323