Finite Dimensional Representations of Quantum Affine Algebras

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.

18 pages

References in corpus (2)

• Affine Toda Solitons and Vertex Operators
• Quantizing SL(N) Solitons and the Hecke Alegbra

Cited by in corpus (4)

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• Automatic Construction of Explicit R Matrices for the One-Parameter Families of Irreducible Typical Highest Weight (0|α) Representations of U_q[gl(m|n)]
• Current Algebraic Structures over Manifolds: Poisson Algebras, q-Deformations and Quantization