Algebraic Lie Theory (07w5025)

Objectives

The objective of the meeting is to bring together people working in the various areas mentioned above, to present the latest developments, and to encourage interchange of ideas and to promote interactive and interdisciplinary collaborations.

More specifically, the groups and algebras which this project seeks to address are algebraic groups, Lie groups, finite reductive groups, reflection groups, Coxeter groups, quantum groups (which are not groups, but Hopf algebras), Hecke algebras and their centralizers. The centralizer of the (Drinfeld) quantum group of classical type acting on tensor space is the Birman-Wenzl algebra which is known to have intimate connections with braids and their topology. Such connections are typical of the area. Lie groups have their origin in real analysis, particularly the study of differential equations invariant under symmetries. They now lie at the heart of several branches of mathematics and form a bridge between apparently diverse fields.

In the last 40 to 50 years, the introduction of the algebraic geometric point of view has extended the range of applicability of ``Lie theoretic'' ideas to ``Lie groups'' over fields other than the real numbers, such as finite or p-adic fields. This has opened up new mathematical worlds, just as the introduction of
``complex numbers'' did in the theory of equations. The major consequence has been cross-fertilisation among several disciplines, the links coming from that part of the representation theory which is independent of the base field. A case in point, particularly relevant to the current proposal, is the example of generalized Gelfand-Graev representations. These were originally defined by Kawanaka in the context of reductive groups over finite fields, but now find applications as far afield as W-algebras in mathematical physics, which can be interpreted as their centralizer algebras, suitably generalized.

On of the new features of the proposal is the unifying role of finite W-algebras which arise, on one hand in Mathematical Physics as so called BRST-quantisations of the Poisson algebras obtained by the generalised Drinfeld-Sokolov reduction, and on the other hand in Algebraic Lie Theory as the endomorphism algebras of the generalised Gelfand--Graev representations. The affine versions of finite W-algebras are prominent in Conformal Fields Theory where they are obtained from Vertex Operator Algebras by means of Quantum Reduction. From the point of view of Lie Theory finite W-algebras are very interesting and linked to non-trivial deformations of the universal enveloping algebras of nilpotent
centralisers in the semisimple Lie algebras. The link with nilpotent orbits and primitive ideals is also very strong here. Recent advances in the theory include work by Kac-Wakimoto on superconformal algebras (the affine case); recent work by Brundan-Kleshchev on the finite W-algebras of type A which established a link with Drinfeld's Yangians; very recent work by Premet on finite W-algebras associated with the minimal nilpotent orbit which established a link with the Joseph ideal.

In summary, the significance of this proposed meeting is that it intends to make serious advances in the core area of Lie group theory and associated representation theory and to explore the interactions between specified fields which are influenced (and vice versa) by these advances.