Quantum Affine Algebras and Universal $R$-Matrix with Spectral Parameter
Using the previous obtained universal $R$-matrix for the quantized nontwisted affine Lie algebras $U_q(A_1^{(1)})$ and $U_q(A_2^{(1)})$, we determine the explicitly spectral-dependent universal $R$-matrix for the corresponding quantum Lie algebras $U_q(A_1)$ and $U_q(A_2)$. As their applications, we reproduce the well-known results in the fundamental representations and we also derive an extreamly explicit formula of the spectral-dependent $R$-matrix for the adjoint representation of $U_q(A_2)$, the simplest non-trival case when the tensor product decomposition of the representation with itself has finite multiplicity.