Two-parameter quantum groups and R-matrices: classical types

We construct finite R-matrices for the first fundamental representation V of two-parameter quantum groups Uₑ, ₒ (g) for classical g, both through the decomposition of V V into irreducibles Uₑ, ₒ (g) -submodules as well as by evaluating the universal R-matrix. The latter is crucially based on the construction of dual PBW-type bases of U^ₑ, ₒ (g) consisting of the ordered products of quantum root vectors defined via (r, s) -bracketings and combinatorics of standard Lyndon words. We further derive explicit formulas for affine R-matrices, both through the Yang-Baxterization technique of M. Ge, Y. Wu, K. Xue, "Explicit trigonometric Yang-Baxterization", Internat. J. Modern Phys. A 6 (1991), no. 21, 3735-3779 and as the unique intertwiner between the tensor product of V (u) and V (v), viewed as modules over two-parameter quantum affine algebras Uₑ, ₒ (g) for classical g. The latter generalizes the formulas of M. Jimbo, "Quantum R matrix for the generalized Toda system", Comm. Math. Phys. 102 (1986), no. 4, 537-547 for one-parametric quantum affine algebras.