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

TL;DR

This paper extends previous work to explicitly construct spectral-dependent universal R-matrices for certain quantum affine algebras and applies them to concrete representations, including a novel explicit formula for the adjoint representation of U_q(A_2).

Contribution

It provides explicit spectral-dependent universal R-matrices for U_q(A_1) and U_q(A_2), including the first explicit formula for the adjoint representation of U_q(A_2).

Findings

Derived spectral-dependent R-matrices for U_q(A_1) and U_q(A_2).

Reproduced known results for fundamental representations.

First explicit formula for the R-matrix of the adjoint representation of U_q(A_2).

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.