# Representation Theory of Finite Dimensional Algebras

### Henning Krause

Universität Bielefeld, Germany### William Crawley-Boevey

University of Leeds, UK### Bernhard Keller

Université Paris Diderot - Paris 7, France### Oeyvind Solberg

University of Trondheim, Norway

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## Abstract

Representation theory of finite dimensional algebras has always been inspired by interactions with other subjects, and Oberwolfach meetings traditionally serve as a forum for such exchange of ideas. The main source of interactions are the many problems in representation theory and in other parts of mathematics which can be formulated in terms of representations of finite dimensional associative algebras. The study of non-semisimple representations took off in the late 20th century with key advances, such as the link to Lie algebras and quantum groups via quivers and Hall algebras, and the use of tilting theory and derived categories to pass from known algebras to new classes of algebras.

In modern work, instead of studying an algebra through its category of representations, or derived category, one may study similar but more general categories. Thus the classification of some classes of hereditary abelian categories or Calabi-Yau triangulated categories fits into this setup. Another recent development, which had just started at the time of the last Oberwolfach meeting in February 2005, and is still being played out, is the interaction with cluster algebras.

At the workshop, there were 46 participants. Among them, there were experts from neighbouring subjects like commutative algebra, algebraic topology, and combinatorics. Compared to previous meetings, the number of participants was reduced, which made it difficult to include representatives of many other fields with close links to representation theory of finite dimensional algebras. What follows is a quick survey of the main themes of the 23 lectures given at the meeting.

\textbf{Cluster combinatorics and Calabi-Yau categories arising from representations of algebras.} Cluster algebras were invented by Fomin and Zelevinsky in 2000 with motivations coming from the study of canonical bases in quantum groups and total positivity in algebraic groups. The combinatorics of these algebras were soon recognized to be closely related to those of tilting theory for hereditary algebras. A collective effort over the last few years has led to a good understanding of these relations for certain classes of cluster algebras. This was made possible by the use of $2$-Calabi-Yau categories constructed from representations of algebras. The introductory talks by Reiten and Iyama were devoted to these developments as well as to the impact of recent important work by Derksen-Weyman-Zelevinsky. In an informal evening presentation, Keller put Derksen-Weyman-Zelevinsky's work into a beautiful homological framework. The talk by Geiss presented cutting-edge results towards the construction of `dual PBW-bases' in large classes of cluster algebras. The proofs are based on subtle techniques from the study of quasi-hereditary algebras, as demonstrated in Schr\"oer's talk. Marsh analyzed fine points of the correspondence between cluster variables and rigid indecomposables and disproved a recent conjecture by Fomin-Zelevinsky. A powerful representation-theoretic model for `higher cluster combinatorics' was presented in the talk by Bin Zhu.

\textbf{Categorification via representations.} The method of categorification has been developed and studied successfully in representation theory by Chuang and Rouquier. They constructed $sl_2$-categorifications for blocks of symmetric groups and used them to establish Brou\'e's abelian defect group conjecture for the symmetric groups. A similar philosophy led to the categorification of cluster algebras via certain $2$-Calabi-Yau categories, where the multiplication in the cluster algebra is modeled by direct sums. A more recent and very promising approach due to Leclerc was presented by Keller. In this case the multiplication is modeled by the tensor product in certain categories of representations of quantum affine algebras. Categorifications also play an important role in low dimensional topology, thanks to important work of Khovanov. This connection was the motivation for Stroppel's talk on convolution algebras arising from Springer fibres.

\textbf{Representation dimension of algebras and complexity of triangulated categories.} The representation dimension of an algebra is a homological invariant which Auslander introduced in 1971 and which remained mysterious for many years thereafter. Some of the modern techniques in representation theory provide now a better understanding. An introductory talk by Ringel discussed the basic ideas and some interesting new phenomena for hereditary algebras. Dimensions of triangulated categories were introduced by Rouquier to obtain lower bounds for representation dimensions and Iyengar presented some new techniques to compute them. The talk of Buchweitz provided a more general perspective for the computation of these dimensions by reviewing the work of Beligiannis and Christensen on projective classes and ghosts in triangulated categories. A description of triangulated structures on additive categories in terms of Hochschild cohomology was presented by Pirashvili.

\textbf{Hereditary categories of geometric origin.} Hereditary categories are in some sense the building blocks for many interesting structures in modern representation theory. Typical examples are categories of coherent sheaves which come equipped with some additional geometric structure. Using this extra structure, Lenzing presented a new description of the stable category of vector bundles on a weighted projective line. The talk of Burban discussed an intriguing connection between vector bundles on elleptic curves and solutions of Yang-Baxter equations. A complete classification of abelian 1-Calabi-Yau categories up to derived equivalence was presented by van Roosmalen.

\textbf{Representations of quivers.} Quivers and their representations have always played a central role in the representation theory of finite dimensional algebras. They provide the link to Lie theory, either through the theorems of Gabriel and Kac, relating possible dimension vectors of indecomposable representations to positive roots, or more directly via Ringel's construction of quantum groups using Ringel-Hall algebras. Progress since the last meeting includes Hausel's announcement of a positive solution of Kac's conjecture that the constant term of the polynomial counting the number of absolutely indecomposable representations over a finite field is the corresponding root multiplicity. Hausel was invited to the meeting, but sadly in the end it was not possible for him to attend. Hausel's result involves hyper-K\"ahler geometry, and in his talk Reineke also used geometry, namely the cohomology of moduli spaces of quiver representations, to prove a formula similar to one conjectured by Kontsevich and Soibelman concerning Donaldson-Thomas type invariants. Chapoton and Hille both gave intriguing talks involving tilting modules for quivers, exceptional sequences and braid group actions. Hubery discussed the connections between Hall algebras and cluster algebras and the existence of Hall polynomials for non-simply laced affine diagrams, using species rather than quivers.

\textbf{Further aspects of algebras and their representations.} Representation theory of finite dimensional algebras has developed immensely since its origin, and it has now, as demonstrated above, profound connections to many other fields. However, the `internal' theory of representation theory is still pushed forward: The talk of Skowro\'nski presented results on algebras with generalized standard almost cyclic coherent Auslander-Reiten components. Representation theory of Lie algebras and algebraic groups is intimately related to finite dimensional algebras which are cellular or quasi-hereditary. These are algebras given by a specific filtration of ideals. K\"onig presented work on how to generalize such a filtration further in order to deal with possibly infinite dimensional building blocks. Benson, Carlson and others developed a theory of support varieties for finitely generated modules over a finite group, and they obtained deep structural information about modular representations of finite groups in terms of the group cohomology ring. These results found their analogous twin results for Lie algebras and Steenrod algebras arising in topology. Similar support varieties have since then been defined for instance for complete intersections, quantum groups and arbitrary finite dimensional algebras. A common denominator for these situations is the presence of a ring of cohomological operations, and in the latter case this is provided by the Hochschild cohomology ring. The talk of Avramov gave an overview over recent results and questions on the Hochschild cohomology ring of an algebra arising in this context. Nakano presented results on the cohomology and support varieties for quantum groups in a quest to find relationships between representations for quantum groups and geometric constructions in complex Lie theory.

The format of the workshop has been a combination of introductory survey lectures and more specialized talks on recent progress. In addition there was plenty of time for informal discussions. Thus the workshop provided an ideal atmosphere for fruitful interaction and exchange of ideas. It is a pleasure to thank the administration and the staff of the Oberwolfach Institute for their efficient support and hospitality.