BIREP – Representations of finite dimensional algebras at Bielefeld

# Seminar Representation Theory, SS 2021

Time and place: Wednesday 10am-noon online

Organisers: Prof. Dr. Henning Krause, Dr. Janina Letz

The seminar covers different topics within the field of representation theory.

### Schedule

• Wednesday, 14 Apr: No meeting
• Wednesday, 21 Apr, 10:15 (online)
• Organizational Meeting
• Raphael Bennett-Tennenhaus: Examples of pp-definable subgroups in the homotopy category of a gentle algebra
In the model theory of modules one considers the language of modules over a fixed ring, and studies a module by means of solutions sets to formulas in this language. So called positive primitive (pp) formulas became of particular importance due to a theorem of Baur in 1976, and their solution sets are called pp-definable subgroups. I will begin by recalling examples of pp-definable subgroups which appear in a classification method from representation theory, known as the functorial filtrations method. I will then attempt to parallel this story with another story, in which one exchanges the category of modules with the homotopy category of complexes of projective modules. To give a concrete comparison between the stories I will fix a running example of a gentle algebra. In a later talk I will explain how this comparison may be used to adapt a classification of so called $\Sigma$-pure-injective objects from the module category to the homotopy category.
Notes
• Wednesday, 28 Apr, 10:15
• Henning Krause: Chase's lemma and its context
Notes
• Friday, 7 May, 13:15
• Raphael Bennett-Tennenhaus: Some descriptions of Sigma-pure-injectivity
In this second of two talks, I will use ideas from the first talk to present two theorems. The first Theorem says that any Sigma-pure-injective module over a string algebra is a coproduct of string and band modules. The second Theorem says that any Sigma-pure-injective object in the homotopy category over a gentle algebra is a coproduct of string and band complexes. I will try to explain some of the tools used in the proofs of these Theorems.
Notes
• Wednesday, 12 May: No meeting
• Wednesday, 19 May, 10:15
• Vincent Klinksiek: A geometric model for the module category of a gentle Algebra by Baur and Simoes
• Wednesday, 26 May, 10:15
• Julia Sauter: Exact structures on exact categories
We give a survey on exact substructures (Quillen-sense) on exact categories and on arbitrary additive categories. Since every additive category admits a unique maximal exact structure, the two tasks are essentially equivalent. But nevertheless, the methods are quite different. For the first task, we look at subfunctors of ${\rm Ext}^1$ (Dräxler-Reiten-Smalo-Solberg and Auslander-Solberg) and for the second we use categories of effaceable functors (Enomoto).
Notes
• Wednesday, 2 Jun, 10:15
• William Crawley-Boevey: Sylvester rank functions for rings and Cohn-Schofield matrix localization (Rank functions, Part I)
I shall discuss Sylvester rank functions for a ring, universal localization of rings, and how they work together. I shall then discuss Schofield's Theorem, giving a 1:1 correspondence between Sylvester rank functions for R and equivalence classes of homomorphisms from R to a simple artinian ring.
Notes
• Wednesday, 9 Jun, 10:15
• Jan-Paul Lerch: Derived localisations of algebras and modules
• Wednesday, 16 Jun, 10:15
• Marc Stephan: Rank functions on triangulated categories and perfect derived categories of rings
I will explain Chuang and Lazarev’s generalization of Sylvester rank functions to triangulated categories.
• Wednesday, 23 Jun, 10:15
• Henning Krause: Derived localization for dg algebras and localizing rank functions (after Chuang and Lazarev)
• Friday, 2 Jul, 13:15
• Janina Letz: Schur algebras of finite type
• Wednesday, 7 Jul, 10:15
• David Fernández: Noncommutative quasi-Poisson geometry and pre-Calabi-Yau algebras
A long-standing problem in Poisson geometry has been to define appropriate “noncommutative Poisson structures”. To solve it, M. Van den Bergh introduced double Poisson algebras and double quasi-Poisson algebras that can be regarded as noncommutative analogues of usual Poisson manifolds and quasi-Poisson manifolds, respectively. Recently, N. Iyudu and M. Kontsevich found an insightful correspondence between double Poisson algebras and pre-Calabi-Yau algebras. The main goal of this talk will be to explain how double quasi-Poisson algebras give rise to pre-Calabi-Yau algebras. Interestingly, they involve an infinite number of nonvanishing higher multiplications weighted by the Bernoulli numbers. This is a joint work with E. Herscovich (Grenoble).
• Wednesday, 14 Jul, 10:15
• Andrew Hubery: Parameterising the tubes for a tame hereditary algebra using noncommutative curves
We will discuss the orbit algebra approach to parameterising the tubes, revealing some beautiful connections between homogeneous normal elements of the orbit algebra, universal extensions, and finite closed subcategories of regular modules. We may also touch upon the connections to universal localizations and generic modules.
• Wednesday, 21 Jul: No meeting