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arXiv · hep-th/9307008

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

Yao-Zhong Zhang·Mark D. Gould
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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 $U_q({\cal G}^{(1)})$ be the quantized nontwisted affine Lie algebra and $U_q({\cal G})$ 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.

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Yao-Zhong Zhang, Mark D. Gould. 1993-07-14. Quantum affine algebras and universal R-matrix with spectral parameter, II. https://arxiv.org/abs/hep-th/9307008

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