Irreducible modules for toroidal Lie algebras
by
S. Eswara Rao
Tata Institute of Fundamental Research

Let G be simple finite dimensional Lie algebra over complex numbers C. The universal central extension of G \otimesC[t₁ ^{+/- 1}, ... , t_n ^{+/- 1}] are called toroidal Lie algebras. For n=1 they are precisely the affine Kac-Moody Lie-algebras so that toroidal Lie-algebras are generalizations of affine Kac-Moody Lie-algebras. The major difference in the toroidal case is that the centre is infinite dimensional unlike the affine case where the centre is one dimensional. This poses a major difficulty in studying representation theory of toroidal Lie algebras. The toroidal Lie-algebras are naturally Zⁿ-graded and there is an infinite dimensional centre with non-zero degree. In a graded irreducible representation the non-zero degree central operator does not act as scalars but acts as a invertible central operators. The main purpose of this paper is to prove that the study of graded irreducible representation for toroidal Lie-algebra is reduced to the study of irreducible representation where the infinite centre acts as scalars.

In the process we prove an interesting result for n >= 2. Let [`(\tau)] be the quotient of \tau by non-zero degree central operators. Then [`(\tau)] does not have modules with finite dimensional weight spaces where centre acts non-trivially.

Date received: June 11, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cals-07.