Matrix Factorization for Solutions of the Yang–Baxter Equation

Abstract

We study solutions of the Yang-Baxter equation on a tensor product of an arbitrary finitedimensional and an arbitrary infinite-dimensional representations of the rank one symmetry algebra. We consider the cases of the Lie algebra sℓ 2 , 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 sℓ 2 , finite-difference operators with trigonometric coefficients in the case of the modular double, or finite-difference operators with coefficients constructed out of 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 formulae.

Higher spin sl_2 R-matrix from equivariant (co)homology

“…The general formula for the rational sl 2 R-matrix in the case of arbitrary spin, written as a function of the Casimir invariant, was obtained as early as in [14] (it is nicely reviewed in the lectures [7]). There is also a representation for the R-matrix in coherent state basis [23], where the R-matrix acts in a tensor product of two finite-dimensional representations, and another one, where the action is via a matrix-valued differential operator in the tensor product of a finite-dimensional representation and a generic Verma module [4]. However these expressions are not explicit enough to be compared with ours.…”