Orthosymplectic -matrices
arXiv:2408.16720 · doi:10.1007/s11005-026-02082-8
Abstract
We present a formula for trigonometric orthosymplectic -matrices associated with any parity sequence, and establish their factorization into the ordered product of -exponents parametrized by positive roots in the corresponding reduced root systems. The latter is crucially based on the construction of orthogonal bases of the positive subalgebra through -bracketings and combinatorics of dominant Lyndon words, as developed in [Clark, Hill, Wang, "Quantum shuffles and quantum supergroups of basic type", Quantum Topol. 7 (2016), no.3, 553-638]. We further evaluate the affine orthosymplectic -matrices, establishing their intertwining property as well as matching them with those obtained through the Yang-Baxterization technique of [Ge, Wu, Xue, "Explicit trigonometric Yang-Baxterization", Internat. J. Modern Phys. A 6 (1991), no.21, 3735-3779]. This reproduces the celebrated formulas of [Jimbo, "Quantum matrix for the generalized Toda system", Comm. Math. Phys. 102 (1986), no.4, 537-547] for classical BCD types and the formula of [Mehta, Dancer, Gould, Links, "Generalized Perk-Schultz models: solutions of the Yang-Baxter equation associated with quantized orthosymplectic superalgebras", J. Phys. A 39 (2006), no.1, 17-26] for the standard parity sequence.
v1: 31pp, comments are welcome! v2: 64pp, significantly improved version (factorization added as a new Section 5; fixed treatment for n=m case adding all details in Appenix B). v3: 64pp, minor corrections, details added