Two-Parameter Quantum Groups and -Matrices: Classical Types - Martin, Ian; Tsymbaliuk, Alexander (Symmetry, Integrability and Geometry: Methods and Applications, 2025)
We construct finite -matrices for the first fundamental representation of two-parameter quantum groups ᵣ,ₛ() for classical , both through the decomposition of ⊗ into irreducibles ᵣ,ₛ()-submodules as well as by evaluating the universal R-matrix. The latter is crucially based on the construction of dual PBW-type bases of ⁺⁻ᵣ,ₛ() consisting of the ordered products of quantum root vectors defined via (, )-bracketings and combinatorics of standard Lyndon words. We further derive explicit formulas for affine -matrices, both through the Yang-Baxterization technique of [Internat. J. Modern Phys. A 6 (1991), 3735-3779] and as the unique intertwiner between the tensor product of (u) and (v), viewed as modules over two-parameter quantum affine algebras ᵣ,ₛ(ˆ) for classical . The latter generalizes the formulas of [Comm. Math. Phys. 102 (1986), 537-547] for one-parametric quantum affine algebras.