BIREP – Representations of finite dimensional algebras at Bielefeld

# Workshop "Representation Theory in Bielefeld - Past and Future" – Abstracts

Thomas Brüstle (Sherbrooke)
Matrix Reductions – Past and Future

Matrix reduction techniques have been one of the important tools in the early days of representation theory. We re-visit some reductions, and propose to re-interpret them as a change of exact structure.

Homological mirror symmetry for non-commutative nodal curves

In my talk, I am going to introduce a class of gentle algebras, which are derived equivalent to certain tame non-commutative nodal curves. Following the approach of Lekili and Polishchuk, this allows to interpret such non-commutative curves as homological mirrors of appropriate graded compact oriented surfaces with non-empty marked boundary.

This is a joint work with Yuriy Drozd.

Ivo Dell'Ambrogio (Lille)
On viewing triangulated categories as rings

The apparently strange idea of viewing triangulated categories as rings, both from the representation theoretic point of view (starting already in the 60's) and from the commutative algebraic point of view (from the 00's), has been surprisingly fruitful and continues to be so to this day. In this talk I will review some of the related notions and results, emphasizing those on the representation-theoretic side and having ties with Bielefeld.

Karin Erdmann (Oxford)
Hybrid algebras

[Joint work with A. Skowroński] We introduce a class of algebras which contains Brauer graph algebras, and weighted surface algebras, and many other tame symmetric algebras, including (most) blocks of finite and tame representation type. This talk discusses the construction, some general properties, and problems.

Steffen Koenig (Stuttgart)
Categories of standardly filtered objects as representations of corings I

Standard objects and categories of objects filtered by these occur frequently in algebra and geometry. As explained by Dlab and Ringel's standardisation theorem, quasi-hereditary algebras provide an appropriate setup for studying such categories. In joint work with Julian Külshammer and Sergiy Ovsienko and with Tomasz Brzezinski and Julian Külshammer, standardly filtered categories have been described as representations of bocses / corings, and applied for instance to the problem of existence of exact Borel subalgebras of quasi-hereditary algebras.

Julian Külshammer (Uppsala)
Categories of standardly filtered objects as representations of corings II

Gabriel's structure theorem asserts that every finite dimensional algebra over an algebraically closed field is Morita equivalent to the quotient of a path algebra of a finite quiver by an admissible ideal. In this correspondence, the quiver is determined by the first extension groups between the simple modules while the number of relations is determined by the second extension group. A new perspective is offered by homotopical algebra, describing the relations as the low degree part of the A-infinity structure on the Ext-algebra of the simples. In this talk, I will explain how the point of view of bocses / corings is analogous to that of a quiver replacing the simple modules in the abelian category of modules by the standard modules, which are the simple objects in the exact category of filtered modules for a quasi-hereditary algebra. This is joint work with Steffen Koenig and Sergiy Ovsienko and with Vanessa Miemietz.

Rosanna Laking (Verona)
Infinite-dimensional pure-injective modules in representation theory

It is well-known that every module over a finite-dimensional algebra $A$ is the direct limit of finite-dimensional modules. It is therefore fair to say that the structure of the category $\mod(A)$ of finite-dimensional modules influences, or even determines, the structure of the whole module category. In many ways the converse is also true: aspects of the structure of certain infinite-dimensional modules (pure-injective modules) determines the structure of $\mod(A)$.

In this talk I will discuss some of the historical aspects of this connection between the infinite-dimensional pure-injective modules and $\mod(A)$, illustrating the discussion by presenting some important classification results. Finally I will describe how new developments in large silting theory fit into this picture by reporting on joint work with Karin Baur.

Gus Lehrer (Sydney)
Invariant theory – classical, quantum and super; recent progress and open problems

There has been an upsurge in activity in invariant theory in recent years, which involves modern methods such as categorification and diagrammatics to address classical problems. I shall outline some new results and open problems.

Yanan Lin (Xiamen)
Weighted projective lines and equivariantizations

The notion of weighted projective lines has been introduced by Geigle-Lenzing in 1987 and gives a geometric treatment to the representation theory of the canonical algebras in the sense of Ringel. Due to Geigle-Lenzing '87, Chen-Chen-Zhou '15, etc., the category of coherent sheaves over a weighted projective line of tubular type is equivalent to the category of equivariant coherent sheaves over some elliptic curve with respect to a certain cyclic group action. Recently, J. Chen, X. Chen and S. Ruan showed that the categories of coherent sheaves over weighted projective lines of tubular type can be related to each other via the equivariantization with respect to certain finite group actions. In particular, the notion of admissible homomorphisms between the string groups of weighted projective lines is introduced which plays an important role to find the group and determine the explicit group action. In this talk, I will introduce the joint work with J. Chen, S. Ruan and H. Zhang which shows how to relate the weighted projective lines via equivariantization by using admissible homomorphisms. As an application, we classify all the coherent sheaves categories over the weighted projective lines of tubular type and of domestic type in the sense of admissible homomorphisms.

Markus Schmidmeier (Boca Raton)
Linear Operators in Representation Theory and Applications

Linear operators and their invariant subspaces are ubiquitous in pure mathematics and in applications. For example, in functional analysis, the "invariant subspace problem" is still open, while in the study of linear time-invariant dynamical systems, the controllable and the non-observable subspaces of the state space are decisive invariant subspaces.

The theory of linear operators and their invariant subspaces, or systems of such subspaces, is intimately linked with Bielefeld representation theory as the Happel-Vossieck list and Ringel's theory of tubular algebras turn out to be indispensable tools.

In my talk I plan to discuss some recent results on linear operators as they provide links to relative homological algebra, the geometry of representation spaces, Young tableau combinatorics and applications.

Jan Schröer (Bonn)
Schemes of modules over gentle algebras

This is joint work with Christof Geiss and Daniel Labardini. We study irreducible components of schemes of modules over gentle algebras. We describe the smooth locus and show that most irreducible components are generically reduced. For the subclass of gentle surface algebras we show that the laminations of an associated marked Riemann surface correspond bijectively to the generically τ-reduced irreducible components. This has applications to the problem of finding bases for cluster algebras.

Sibylle Schroll (Leicester)
Geometric Models and Derived Invariants for Gentle Algebras

Gentle algebras are a class of well-studied tame algebras which originate in the context of iterated tilting in the 1980’s. Since then they have made an appearance in various different settings such as categorifications of cluster algebras, N=2 gauge theories and homological mirror symmetry of surfaces. In this talk, we will describe a geometric model of the bounded derived category of a gentle algebra and we will show how this model relates to the (partially wrapped) Fukaya category of a surface with boundary and stops. We will use the geometric model to give a complete derived invariant for gentle algebras. (This is talk is based on joint work with S. Opper and P.-G. Plamondon and joint work with C. Amiot and P.-G. Plamondon).

Andrzej Skowroński (Toruń)
Algebras of generalized quaternion type

[Joint work with K. Erdmann] Periodic algebras form a prominent class of selfinjective algebras, having interesting connections with group theory, topology, singularity theory and cluster algebras. The talk concerns the classification of all periodic symmetric tame algebras over an algebraically closed field. I will concentrate mostly on the classification of algebras of generalized quaternion type, which seem to provide all periodic symmetric tame algebras of non-polynomial growth.

Pu Zhang (Shanghai)
Gorenstein-projective and semi-Gorenstein-projective modules

Let $A$ be an artin algebra. An $A$-module $M$ will be said to be semi-Gorenstein-projective provided that $\Ext^i(M,A) = 0$ for all $i>0$. All Gorenstein-projective modules are semi-Gorenstein-projective and only few and quite complicated examples of semi-Gorenstein-projective modules which are not Gorenstein-projective have been known. One of the aims of this talk is to provide conditions on $A$ such that all semi-Gorenstein-projective left modules are Gorenstein-projective. In particular, this is the case if there are only finitely many isomorphism classes of indecomposable left modules which are both semi-Gorenstein-projective and torsionless. On the other hand, we exhibit a 6-dimensional algebra with a semi-Gorenstein-projective module which is not torsionless (thus not Gorenstein-projective). Actually, also its $A$-dual module is semi-Gorenstein-projective. In this way, we show the independence of the total reflexivity conditions of Avramov and Martsinkovsky, thus completing a partial proof by Jorgensen and Sega.

This is a joint work with Claus Michael Ringel.