Representations of Affine Quantum Function Algebras

Abstract

Let $C$ be a symmetrizable generalized Cartan Matrix, and $q$ an indeterminate. ${\fg}(C)$ is the Kac-Moody Lie algebra and $U=U_q({\fg}(C))$ the associated quantum enveloping algebra over $k=\mathbb{Q}(q)$. The quantum function algebra ${\mathbb{C}}_q[G]$ is defined as a suitable $U$-bisubalgebra of the dual space ${\mathrm{hom}}_k(U,k)$ which can be described using matrix elements of integrable $U$-modules. For $\fg$ affine, the highest weight modules of $C_q[G]$ are constructed and, assuming a minimality condition, their (unitarizable) irreducible quotients are shown to be in a 1-1 correspondence with the reduced elements of the Weyl group of $\mathfrak{g}(C)$. Further, these simple module are described in terms of the $C_q[S{L}_2]$-modules obtained by restriction, and they satisfy a Tensor Product theorem, similar to the finite type case.

31 pages, adapted from PhD thesis, May 2002, KSU