On the Construction of Trigonometric Solutions of the Yang-Baxter Equation

We describe the construction of trigonometric R-matrices corresponding to the (multiplicity-free) tensor product of any two irreducible representations of a quantum algebra $U_q(\G)$. Our method is a generalization of the tensor product graph method to the case of two different representations. It yields the decomposition of the R-matrix into projection operators. Many new examples of trigonometric R-matrices (solutions to the spectral parameter dependent Yang-Baxter equation) are constructed using this approach.