Finite Dimensional Representations of Quantum Affine Algebras

TL;DR

This paper presents a general method for constructing finite dimensional representations of quantum affine algebras, highlighting differences from classical cases and illustrating with explicit examples relevant to affine Toda theory.

Contribution

It introduces a new construction technique for finite dimensional modules of quantum affine algebras, especially when classical decompositions do not directly extend.

Findings

Explicit constructions for $ ilde{ ext{C}}_2$ and $ ilde{ ext{G}}_2$ cases.

Finite dimensional modules relate to soliton multiplet structures.

Demonstrates how quantum deformation alters classical module structures.

Abstract

We give a general construction for finite dimensional representations of $U_q(\hat{\G})$ where $\hat{\G}$ is a non-twisted affine Kac-Moody algebra with no derivation and zero central charge. At $q=1$ this is trivial because $U(\hat{\G})=U({\G})\otimes \C(x,x^{-1})$ with $\G$ a finite dimensional Lie algebra. But this fact no longer holds after quantum deformation. In most cases it is necessary to take the direct sum of several irreducible $U_q({\G})$-modules to form an irreducible $U_q(\hat{\G})$-module which becomes reducible at $q = 1$. We illustrate our technique by working out explicit examples for $\hat{\G}=\hat{C}_2$ and $\hat{\G}=\hat{G}_2$. These finite dimensional modules determine the multiplet structure of solitons in affine Toda theory.