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 finite-dimensional 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.

References in corpus (9)

• Discrete Heisenberg-Weyl Group and Modular Group
• Quantization of models with non-compact quantum group symmetry. Modular XXZ magnet and lattice sinh-Gordon model
• Faddeev-Volkov solution of the Yang-Baxter Equation and Discrete Conformal Symmetry
• Yang-Baxter R operators and parameter permutations
• The universal R-matrix and factorization of the L-operators related to the Baxter Q-operators
• New elliptic solutions of the Yang-Baxter equation
• From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation
• Vacuum curves of elliptic L-operators and representations of Sklyanin algebra
• Factorization of the R-matrix.II