BIREP – Representations of finite dimensional algebras at Bielefeld

# Workshop "Representations of Algebras and Sheaves"

### Saturday 13 November 2021

Location: Zentrum für interdisziplinäre Forschung (ZiF) in Bielefeld

Participation: This workshop is only for invited participants because of restrictions due to the pandemic.

The current Corona Protection Ordinance applies, in particular the "3G rule": participation is only possible for vaccinated (geimpft), tested (getestet) or recovered (genesen) persons. Participants are further required to wear a medical mask indoors at all times except when at their fixed seats.

Organisers: Bill Crawley-Boevey and Henning Krause

# Workshop Programme

### Saturday 13 November

 09:00 – 10:00 Teresa Conde (Stuttgart)Rank functions on triangulated categories via abelianisation Coffee break 10:30 – 11:30 Raphael Bennett-Tennenhaus (Bielefeld)On the notion of an extriangulated functor 11:30 – 12:30 Klaus Bongartz (Wuppertal)Representation-finite selfinjective algebras, coverings, multiplicative bases ... Lunch break 13:30 – 14:30 Magdalena Boos (Bochum)Symmetric quiver representations and degenerations Coffee break 15:00 – 16:00 René Marczinzik (Stuttgart)Auslander regular algebras and generalisations of higher Auslander algebras 16:00 – 17:00 Gustavo Jasso (Bonn)Generalised BGP reflection functors

# Talks and Abstracts

### Raphael Bennett-Tennenhaus (Bielefeld)On the notion of an extriangulated functor

The definition of an extriangulated category was given by Nakaoka and Palu. This notion simultaneously generalises those of an exact category and a triangulated category. Examples of extriangulated categories which need not be exact nor triangulated appear, for example, when studying pure-exact sequences in compactly generated triangulated categories.

In this talk I will discuss what are called extriangulated functors, from joint work with myself and Amit Shah. As above, the definition unifies that of exact functors between exact categories, and triangulated functors between triangulated categories. Other examples of extriangulated functors include the delta functor from an exact category to its derived category, and the restricted Yoneda functor from a compactly generated triangulated category to the appropriate functor category. Time permitted, I will explain the role of a category of extensions.

### Klaus Bongartz (Wuppertal)Representation-finite selfinjective algebras, coverings, multiplicative bases ...

An 'easy' complete proof for the classification of the representation-finite selfinjective algebras including the non-standard ones is sketched. The intimate relations to further developments are also discussed, some points in the literature criticized, some errors corrected. Details to all this are in a forthcoming article.

### Magdalena Boos (Bochum)Symmetric quiver representations and degenerations

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. We are interested in the orbits and their closures of said actions. Orbit closure relations lead us to considering symmetric degenerations for which one of the most important questions is: are symmetric degenerations induced by "usual" degenerations in the representation variety of the underlying quiver? We look at (counter)examples and recent results. This is joint work with G. Cerulli Irelli.

### Teresa Conde (Stuttgart)Rank functions on triangulated categories via abelianisation

Motivated by the work of Cohn and Schofield on Sylvester rank functions on rings, Chuang and Lazarev have recently introduced the notion of a rank function on a triangulated category. A rank function is a nonnegative real-valued, additive, translation-invariant function on the objects of a triangulated category for which the triangle inequality on distinguished triangles holds. It turns out that a rank function on $C$ can be recast as translation-invariant additive function on its abelianisation $\mathrm{mod}{-}C$. This allows us to relate integral-valued rank functions on $C$ with endofinite cohomological functors on $C$ and, when $C$ is the subcategory of compact objects of a compactly generated triangulated category $T$, with endofinite objects in $T$ and certain closed sets of the Ziegler spectrum of $T$.

This talk is based on joint work in progress with Mikhail Gorsky, Frederik Marks and Alexandra Zvonareva.

### Gustavo Jasso (Bonn)Generalised BGP reflection functors

The derived BGP reflection functors are among the simplest kind of derived equivalences. I will explain how to leverage the gluing operations afforded by the powerful language of infinity-category theory to obtain vast generalisations of these equivalences which comprise, in particular, earlier equivalences constructed by Ladkani and by Rahn and Stoviceck. Knowledge of infinity-category theory will not be assumed. The talk is based on joint work with Tobias Dyckerhoff and Tashi Walde.

### René Marczinzik (Stuttgart)Auslander regular algebras and generalisations of higher Auslander algebras

We discuss recent results related to Auslander regular algebras and partially ordered sets. We then define locally higher Auslander algebras and minimal Auslander-Cohen-Macaulay algebras as a generalisation of higher Auslander algebras and see how those new classes of algebras can help to answer an open question by Green and another open question by Auslander and Reiten.