Workshop "Discrete Categories in Representation Theory" – Abstracts

Andre Beineke (Bielefeld)
Δ-critical quasi-hereditary algebras

Happel and Vossieck classified the tame concealed algebras as the path algebras of quivers of frame types given by the extended Dynkin diagramms. Ringel then defined Δ-critical algebras as the tame concealed quasi-hereditary algebras for which all costandard modules are preinjective.

Any Δ-critical algebra A (with standard modules Δ and costandard modules ∇) is in particular quasi-hereditary, and Ringel showed that in this case the category of A-modules with a Δ-filtration is a functorially finite subcategory of A-mod which is closed under extensions and direct summands, and which according to Auslander and Smalø then has (relative) AR-sequences. Further on, since all costandard modules are preinjective, the algebra's characteristic tilting module T (which has a Δ- and a ∇-filtration) is preinjective, too, and the category of A-modules with a Δ-filtration contains in particular all preprojective and all regular A-modules. It has a preprojective component of type A and a preinjective component of type B, where B is the Ringel dual B=End(T)op.

So Δ-critical algebras yield examples of categories of modules which have a preprojective and a preinjective component, but not necessarily of the same type. In my talk, I will give an overview of the many examples that turn up this way for the extended Dynkin diagramms of type E, and explain some ways of making the long list of Δ-critical algebras more accessible. For example considering frame types instead of individual algebras or quivers, just like Happel and Vossieck did, can be helpful.

Grzegorz Bobinski (Torun)
Derived Hall algebras for derived discrete algebras

Hall algebras were introduced into representation theory by Ringel and played an important role due their connections with Lie theory and quantum groups. The construction of these Hall algebras was based on an abelian structure of module categories. Toën presented a Hall algebra construction for triangulated categories. These Hall algebras were studied and described in some contexts, for example by Keller, Yong and Zhou for the triangulated categories generated by a spherical object. In the talk I will discuss the case of triangulated categories associated with derived discrete algebras. This is a report on a joint work with Janusz Schmude.

Nathan Broomhead (Plymouth)
Thick subcategories of discrete derived categories

I will explain some work, in which I describe the lattices of thick subcategories of discrete derived categories. This is done using certain generating collections of exceptional and sphere-like objects related to non-crossing configurations of arcs in a geometric model.

Frédéric Chapoton (Strasbourg)
On some representation-theoretic aspects of Tamari lattices

Tamari lattices are nice combinatorial partial orders, at the crossroad of algebraic topology and several recent directions in representation theory, including cluster algebras and tau-tilting theory. I will present various results and conjectures on these posets and their relatives, namely

  • the conjectural property of being fractionally-Calabi-Yau, that should extend to all cambrian lattices but not to more general partial orders on tau-tilting objects,
  • the study of representation types of cambrian lattices, most of them being wild,
  • an equivalence of derived categories between two algebras describing intervals in a poset, initially motivated by the case of the Tamari lattice, but holding in full generality.

The last two points are joint work with Baptiste Rognerud.

Christof Geiss (Mexico/Bonn)
A reminder on derived tame algebras and clans

In this talk I review some of my work which was done at the end of the last century. Clans are a class of tame matrix problems which were extensively studied by W. Crawley-Boevey. Similar problems were studied under the names "bundles of semi-chains" and bushes by Bondarenko resp. Bangming Deng. The classification of the indecomposable representations is in terms of certain words (symmetric resp. asymmetric strings and bands) and is essentially independent of the ground field. We showed that the description of irreducible homomorphisms between those representations is also independent of the ground field.

If the characteristic of the ground field is not 2, clannish algebras can be viewed as skew group algebras of string algebras, which gives, together with the above result a cheap way to describe their AR-quiver for any base field. In joint work with J.A. de la Peña we introduced a particular class of clannish algebras, the so-called skew gentle algebras. As their name suggests, if the characteristic of the ground field is not 2, they are skew group algebras of certain gentle algebras. In any case, their repetitive algebras are, up to an element in the socle, clannish, thus they are derived tame, and we can describe the stable AR-quiver quite well. In contrast to gentle algebras, the class of skew gentle algebras is not closed under derived equivalence.

Let now be k an algebraically closed field. We say that a finite dimensional k-algebras is weakly simply connected if it can be obtained from the ground field by a sequence of on-point extensions of co-extensions with indecomposable modules. We show, that a weakly connected, derived tame algebra is either piecewise hereditary, or it is derived equivalent to a skew gentle algebra. In the latter case, the derived equivalence class is determined by the number of vertices and the corank of the Euler form, which is non-negative. Moreover, a tree algebras is derived tame, if and only if its Euler form is non-negative.

Bernhard Keller (Paris)
On green sequences

In this mainly expository talk, we will report on green sequences, from their invention motivated by quantum dilogarithm identities respectively BPS state counting to recent beautiful work by Igusa.

Rosanna Laking (Bonn)
Discrete categories and indecomposable objects

Indecomposable objects are considered to be the fundamental building blocks for many categories arising in representation theory. For example, the category of finite-dimensional modules over an algebra is a Krull-Schmidt category. In this setting, it makes sense to aim to classify the indecomposable objects in order to understand the category as a whole. However, this approach does not often extend to the categories where ‘large’ objects exist. For example, there are infinite-dimensional modules over finite-dimensional algebras with no indecomposable direct summands.

In this talk we will discuss a recurring theme in the study of discrete categories: the property that every object, even if it is ‘large’, has an indecomposable pure-injective direct summand. Our examples will include uniserial abelian categories (studied by Krause and Vossieck) and the homotopy category of projective modules over a derived-discrete algebra (studied in joint work with Arnesen, Pauksztello and Prest).

Steffen Oppermann (Trondheim)
Higher dimensional Auslander-Reiten components

This talk is based on joint work with Martin Herschend.

In classical Auslander-Reiten theory, one first defines almost split sequence, and subsequently arranges them unto AR components.

In higher dimensional Auslander-Reiten theory the notion of almost split sequences depends on the choice of a rigid (hence discrete) subcategory. In this setup, we suggest an axiomatic definition of AR components, and explore examples and properties.

David Pauksztello (Lancaster)
Discrete triangulated categories

The notion of a discrete derived category was first introduced by Vossieck, who classified the algebras admitting such a derived category. Due to their tangible nature, discrete derived categories provide a natural laboratory in which to study concretely many aspects of homological algebra. Unfortunately, Vossieck's definition hinges on the existence of a bounded t-structure, which some triangulated categories do not possess. Examples include triangulated categories generated by 'negative spherical objects', which occur in the context of higher cluster categories of type A infinity. In this talk, we compare and contrast different aspects of discrete triangulated categories with a view toward a good working definition of such a category. This is a report on joint work with Nathan Broomhead and David Ploog.

Claus Michael Ringel (Bielefeld)
Pre-Dinner-Lecture: Dieter Vossieck and the development of the representation theory of Artin algebras

This workshop is organized in order to celebrate the 60th birthday of Dieter Vossieck: his celebrated paper "The algebras with discrete derived category" has to be seen as the starting point of a development which is discussed in this workshop.

Dieter Vossieck has published only few papers, but his influence is much larger. We will outline some of these contributions.

Alexandra Zvonareva (St. Petersburg)
t-structures and the stability manifold for silting-discrete algebras

In this talk I will present joint work with David Pauksztello and Manuel Saorín. We show that any bounded t-structure in the bounded derived category of a silting-discrete algebra is algebraic, i.e. has a length heart with finitely many simple objects. As a corollary, we obtain that the space of Bridgeland stability conditions for a silting-discrete algebra is contractible.