Quantum affine algebras and universal R-matrix with spectral parameter, II

Yao-Zhong Zhang; Mark D. Gould

paper

Quantum affine algebras and universal R-matrix with spectral parameter, II

arXiv:hep-th/9307008

Abstract

This paper is an extended version of our previous short letter \cite{ZG2} and is attempted to give a detailed account for the results presented in that paper. Let $U_{q} (G^{(1)})$ be the quantized nontwisted affine Lie algebra and $U_{q} (G)$ be the corresponding quantum simple Lie algebra. Using the previous obtained universal $R$ -matrix for $U_{q} (A_{1}^{(1)})$ and $U_{q} (A_{2}^{(1)})$ , we determine the explicitly spectral-dependent universal $R$ -matrix for $U_{q} (A_{1})$ and $U_{q} (A_{2})$ . We apply these spectral-dependent universal $R$ -matrix to some concrete representations. We then reproduce the well-known results for the fundamental representations and we are also able to derive for the first time the extreamly explicit and compact formula of the spectral-dependent $R$ -matrix for the adjoint representation of $U_{q} (A_{2})$ , the simplest nontrival case when the tensor product of the representations is {\em not} multiplicity-free.

22 pages