Interaction of Finite Dimensional Algebras with other areas of Mathematics

The Workshop will concentrate on several topics reflecting a close relationship between the theory of finite dimensional associative algebras and other areas of Mathematics. It will deal, in particular, with relations between Lie theory and the representation theory of quivers. Methods concerning quivers and their representations have been used in the past 30 years extensively in order to describe the structure of length categories (abelian categories where every object has a finite composition series) which arise very frequently not only in algebra, but also in geometry and analysis. They enable a better understanding of the indecomposable objects and allow often a definite presentation of the category by generators and relations. There are several quite surprising relations to Lie theory: first of all, several length categories play a prominent role in Lie theory such as the category $\Cal O$ of categories of Harish-Chandra modules and they can be investigated successfully using the representation theory of finite dimensional algebras. And second, one may use the representation theory of special finite dimensional algebras in order to construct Lie algebras and quantum groups.

The Workshop will also reflect the latest work on quasi-hereditary algebras and their generalizations. These algebras may be viewed as generalizations of quiver algebras more appropriate for understanding the standard module/irreducible module relationship in the category$\Cal O$, perverse sheaves, and categories of representations of algebraic groups in positive characteristic. They were introduced by Cline-Parshall-Scott in their efforts to understand Kazhdan-Lusztig theory, and notably developed further by work of Dlab and Ringel, both jointly and individually. Some of the most interesting generalizations of quasi-hereditary algebras, called stratified algebras, have been defined in independent (and slightly different) ways by Agoston-Dlab-Lukács and Cline-Parshall-Scott. In each case, the stratification involved occurs categorically at the derived category level and internally in terms of particular ideals. Applications include nondescribing characteristic representations of finite groups of Lie type (Du-Parshall-Scott), to pure finite group representation theory (Webb), and to new generalizations, involving non-highest weight irreducible modules, of characteristic 0 Lie algebra representation theory (Futorny-Koenig-Mazorchuk).

In other recent developments, Soergel and others have continued to push the model of finite dimensional modules of cohomology rings as a replacement for relevant perverse sheaf categories, and Ginzburg et al have announced a re-working of the Kazhdan-Lusztig + Kashiwara - Tanisaki understanding of quantum group representations at an lth root of 1 in these terms. Tilting modules, an older development originating in quiver theory, now impact many areas of representation theory, especially characteristic p, through work of Ringel, Donkin, Soergel and Andersen, for example, and have even proved useful for the study of maximal subgroups of finite groups of Lie type (Seitz, Saxl).

Also, Schur-Weyl duality that was at the heart of the introduction of quantum groups by Drinfeld and Jimbo is now understood completely in a characteristic-free quantum context (Donkin, Du- Parshall-Scott) with tilting modules as a major tool. Vesserot, generalizing work of Erdmann, has used this and Ariki's work mentioned below to provide a complete equivalence of the problems of understanding representations of Hecke Algebras representations in type A and corresponding representations of q-Schur algebras. The latter, introduced by Jimbo for physics and, independently, introduced by Dipper-James for finite groups of Lie type in nondescribing characteristic, control finite rank representations of quantum enveloping algebras.

The workshop will also focus the attention to the use of the representation theory of special finite dimensional algebras in order to construct Lie algebras and quantum groups. The exciting development in the representation theory of finite dimensional algebras in the last 30 years was based on the use of very intricate combinatorial methods (quivers, root systems, posets, integral quadratic forms). The combinatorial approach has an algebraic counterpart which leads to Hall polynomials and quantum groups. Presently, the investigations use machinery of perverse sheaves (Lusztig), methods of differential geometry (Nakajima) and Hall algebras (Ringel).

An exciting development has been the realization (by Jie Xiao and others) of non-affine Kac-Moody Lie algebras in natural terms, not just using abstract generators and relations (an unsolved problem for many years). In the affine case, especially, through the Fock space representation in type A, many new connections with known finite dimensional or finite rank Hecke algebras have been discovered by Ariki, who also proved a conjecture of Lascoux-Leclerc-Thibon in type A, and in other classical types (B and C, extended to type D by Jun Hu) successfully parameterized positive characteristic irreducible representations.Older work of Kleshchev on the modular representation of the symmetric group, proving a conjecture of Millineux, is now seen as having broader significance in Jun Hu's work.

The main interest at present lies in an extension of these investigations to Lie algebras defined by intersection matrices or even more general data. It has been outlined already that some of the elliptical Lie algebras studied by Saito (those of type D_4, E_6, E_7, E_8) can be obtained using tubular algebras and it will be of interest to deal with those Lie algebras which arise from a general canonical algebra (or, equivalently, a weighted projective line).

All the references to "representation theory" above should be interpreted in the broadest current sense, which includes homological as well as geometrical considerations. Note that representations of finite dimensional algebras, through translation to quiver or other problems, often enter into other geometric issues beyond those strictly related to Lie representation theory. Though it is not a principal focus of our Workshop, the connections of representation theory as specific to Lie theory and Lie groups with the general theory of representations of finite dimensional algebras and of finite groups is always present in our thinking. The unification of as much as possible of Mathematics in the areas of these disciplines is an additional goal.

Thus, the scope of the applications of finite dimensional algebras will extend to algebraic geometry, automorphic forms, finite group representations and mathematical physics along the lines initiated at the 1992 Annual Canadian Mathematical Seminar at Carleton University and published by Kluwer Academic Publishers as Volume 424 of Series C: Mathematical and Physical Sciences. It should be pointed out that extensive work is in progress and that significant fresh developments are expected in these areas by 2004.

The timeliness of our proposal should be clear from the above developments. It is easy to summarize some of the remaining objectives: 1) Understanding in the broadest infinite-dimensional terms, but through finite dimensional algebras, the representations of Lie algebras in characteristic 0, and related geometric structures, such as perverse sheaves. 2) Understanding representations finite groups of Lie type in characteristic p, and related theory for the symmetric groups and other Coxeter groups important in finite groups and Lie theory. 3) The use of the representation theory of quivers and species for getting insight into the structure of Kac-Moody Lie algebras which are not of finite or affine type. 4) The use of tubular algebras in order to get insight into the structure of the so-called elliptical Lie algebras.

In addition to people mentioned explicitly above, the following names should be mentioned as potential participants - speakers. Mathas, Geck and Hiss to report on Hecke algebras and algebraic groups. Possibly also Rouquier, Broué and Rickard, as well as Putcha and Renner whose work could have been mentioned above. The same applies to Buchweitz, Crawley-Boevey and Reiten who should present the latest developments in the representation theory of quivers and its interaction with geometry. In addition, there is a quite large number of more junior mathematicians (Gruber, Brundan, Rui, Hodge, Francis to name a few) who work in the area of the Workshop and will be interested in participating; a more detailed list of them may be compiled later.

Confirmed Participants

Schedule (pdf file)