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

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This paper is an extended version of our previous short letter ZG2 and is attempted to give a detailed account for the results presented in that paper. Let Uq(G⁽¹⁾) be the quantized nontwisted affine Lie algebra and Uq(G) be the corresponding quantum simple Lie algebra. Using the previous obtained universal R-matrix for Uq(A₁⁽¹⁾) and Uq(A₂⁽¹⁾), we determine the explicitly spectral-dependent universal R-matrix for Uq(A₁) and Uq(A₂). 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 Uq(A₂), the simplest nontrival case when the tensor product of the representations is not multiplicity-free.