Matrix Factorization for Solutions of the Yang–Baxter Equation

Abstract

We study solutions of the Yang–Baxter equation on the tensor product of an arbitrary finite-dimensional and an arbitrary infinite-dimensional representations of rank 1 symmetry algebra. We consider the cases of the Lie algebra sℓ₂, the modular double (trigonometric deformation), and the Sklyanin algebra (elliptic deformation). The solutions are matrices with operator entries. The matrix elements are differential operators in the case of sℓ₂, finite-difference operators with trigonometric coefficients in the case of the modular double, or finite-difference operators with coefficients constructed of the Jacobi theta functions in the case of the Sklyanin algebra. We find a new factorized form of the rational, trigonometric, and elliptic solutions, which drastically simplifies them. We show that they are products of several simply organized matrices and obtain for them explicit formulas. Bibliography: 44 titles.