Unitarity and Complete Reducibility of Certain Modules over Quantized Affine Lie Algebras

TL;DR

This paper proves that all integrable representations of certain quantized affine Lie algebras are completely reducible when the deformation parameter is not a root of unity, and classifies unitary modules over these algebras.

Contribution

It establishes complete reducibility of integrable modules over quantized affine Lie algebras and characterizes unitary modules for positive q.

Findings

All integrable representations in the category are completely reducible.

Every integrable irreducible highest weight module over the nontwisted algebra with q>0 is unitary.

The results hold when q is not a root of unity.

Abstract

Let $U_q(\hat{\cal G})$ denote the quantized affine Lie algebra and $U_q({\cal G}^{(1)})$ the quantized {\em nontwisted} affine Lie algebra. Let ${\cal O}_{\rm fin}$ be the category defined in section 3. We show that when the deformation parameter $q$ is not a root of unit all integrable representations of $U_q(\hat{\cal G})$ in the category ${\cal O}_{\rm fin}$ are completely reducible and that every integrable irreducible highest weight module over $U_q({\cal G}^{(1)})$ corresponding to $q>0$ is equivalent to a unitary module.