The Algebras and Representation Theory Network ARTIG aims to promote exchange and regular meetings between the research groups at the universities of

Aachen - Bielefeld - Bochum - Bonn - Hannover - Kaiserslautern - Köln - Münster - Paderborn - Stuttgart

ARTIG Homepage here

1st Meeting: 27-28 January (University of Cologne)      

Speakers: 

Flashtalks:

Schedule

Friday:

Benedikt Fluhr (TU Muenchen)

Janina Letz (Bielefeld)

Jonas Nehme (Bonn)

Aran Tattar (Köln)

Saturday:

Location: Mathematical Institute,
University of Cologne,
Weyertal 86-90,
50931 Köln 

All talks will take place in the lecture theatre on the second floor. Registration and coffee breaks will be in Seminar Room 2 right next to the lecture theatre. 

Titles & Abstracts

Magdalena Boos (Bochum): On symmetric quiver representations

Abstract: The notion of a symmetric quiver was first introduced by Derksen and Weyman in 2002. Symmetric quiver representations are collected in so-called symmetric representation varieties which are acted on by reductive groups via change of basis. They can be seen as classical type B, C and D analogues of quiver representation varieties. We give an introduction to symmetric quiver representations, motivate our interest in the theory and show first results on the mentioned group actions. This is joint work with G. Cerulli Irelli.


Igor Burban (Paderborn): Hall algebra of a half ruled regular non-commutative curve 

Abstract: The notion of an exceptional curve was introduced by Lenzing in 1998. These are non-commutative hereditary projective curves admitting a tilting object. Weighted projective lines are basic examples of such curves. However, the class of exceptional curves is not exhausted by weighted projective lines when the ground field is not supposed to be algebraically closed.

In my talk I shall discuss a specific example of an exceptional curve over a finite field, which is half ruled regular in the sense of Artin and de Jong. I shall prove that the composition Hall algebra of the corresponding category of coherent sheaves can be identified with the Drinfeld realization of the quantum affine algebra of type A_2^2.

This is a joint work in progress with Heike Herr.


Steffen König (Stuttgart): Rigidity dimension of algebras

Abstract: When considering a self-injective algebra as "singular", one may want to resolve it by a "regular" algebra of finite global dimension; one may think of a Schur algebra resolving a group algebra of a symmetric group or an Auslander algebra resolving an algebra of finite representation type. Moreover, one may want to measure the quality of any such resolution or of the best possible one. Rigidity dimension is a homological dimension that can be used to formulate such questions in precise terms. Basic questions are about finiteness and about invariance under derived or stable equivalences.

This is joint work with Hongxing Chen, Ming Fang, Otto Kerner and Kunio Yamagata.


Julia Sauter (Bielefeld): Tilting theory in exact categories

Abstract: We explore tilting theory for arbitrary exact categories. With various extra assumptions you find this in many places - but no treatment for general exact categories which unifies them.

The main open question is when do they induce triangle equivalences on bounded derived categories (and of course on which derived categories)?

I choose a specific functor to a functor category over the tilting subcategory (with enough projectives) because it seems to be a natural generalization of our well-known tilting situation (tilting modules over artin algebras). My main result is for exact categories with enough projectives: They induce derived equivalences if and only if they are finitely resolving subcategories in an arbitrary functor category with enough projectives. At the end I talk about examples.


Ulrich Thiel (Kaiserslautern): The rank one property for free Frobenius extensions

Abstract: The Cartan matrix of a finite-dimensional algebra is the matrix of multiplicities of simple modules in indecomposable projective modules. This is crucial information about the representation theory of the algebra. In my talk I will present a general setting including several important examples from Lie theory, such as restricted quantized enveloping algebras at roots of unity, in which we could prove that the Cartan matrix has the remarkable property of being blockwise of rank one.

This is joint work with Gwyn Bellamy. [Slides]

Flashtalks

Benedikt Fluhr (TU München): A Sheaf-Theoretical Happel Functor

Abstract: Dieter Happel provided an equivalence between the mesh category of the repetition quiver of a Dynkin quiver Q and the full subcategory of indecomposables of the derived category of Q. In particular, Happel showed that for each n ∈ ℕ the derived categories of all Aₙ-quivers areequivalent. We provide a continuous counterpart of this result, with certain topological spaces including the real numbers taking on the role of Aₙ-quivers and sheaves with finite-dimensional cohomology on connected opens taking on the role of representations. This is joint work with Ulrich Bauer (https://arxiv.org/abs/2205.15275) and closely related to a research article by Igusa, Rock, and Todorov (https://arxiv.org/abs/1909.10499).  [Slides, works best in Google Chrome]


Janina Letz (Bielefeld): Brown representability for triangulated categories with a linear action by a graded ring

Statements that give necessary and sufficient conditions for a functor to be representable are called Brown representability. For triangulated categories such results were established by Neeman, Bondal and van den Bergh, and Rouquier. The latter three established results for strongly generated triangulated categories with a finiteness condition over a ring. I will present a generalization to triangulated categories with a finiteness condition over a graded ring and give some necessary and sufficient conditions for a functor to be graded representable, that is isomorphic to the coproduct of the Hom functors taken over all shifts.  [Slides]


Jonas Nehme (Bonn): A Khovanov algebra for the periplectic Lie superalgebra

Abstract: Khovanov algebras provide strong tools for understanding the category of finite dimensional representations of a Lie superalgebra. They give a diagrammatic description of the endomorphism ring of a projective generator. I will talk about the Khovanov algebra for the periplectic Lie superalgebra and present some results on the action of translation functors on finite dimensional $\mathfrak{p}(n)$-modules using this algebra. [Slides]


Aran Tattar (Köln): Chains of torsion classes and weak stability conditions

Abstract: Based on joint work-in-progress with Hipolito Treffinger. Joyce introduced the concept of weak stability conditions for an abelian category as a generalisation of Rudakov's stability conditions. In this talk, we show an explicit relation between chains of torsion classes and weak stability conditions in an abelian category and  discuss the structure of the space of chains of torsion classes. [Slides]

Dinner on Friday

Our informal conference dinner will be held at "Bierhaus am Rhein" at 18:30. 

We will travel there together after the last talk on Friday.

Bierhaus am Rhein

Frankenwerft 27

50667 Köln

https://bierhaus-am-rhein.de/

This meeting is organised by Henning Krause (University Bielefeld), Sibylle Schroll (University of Cologne/NTNU) and Catharina Stroppel (University Bonn) with assistance from Maximilian Kaipel (University of Cologne).