Quantum Yang-Baxter Equation, Braided Semigroups, and Dynamical Yang-Baxter Maps

Abstract

We prove that braided semigroups with suitable conditions can produce solutions to the quantum Yang-Baxter equation in every tensor category. As an application, some dynamical Yang-Baxter maps, set-theoretic solutions to a version of the quantum dynamical Yang-Baxter equation, are constructed.

Construction of Hopf algebroids

“…Motivated by the FRT construction of a q-analogue of the function space [7,8], one of the author constructed Hopf algebroids A σ [15,32,33] by means of dynamical Yang-Baxter maps σ [14,18,19,20,25,28,29,30,31]. This σ is a set-theoretic solution to a version of the quantum dynamical Yang-Baxter equation [9,10], a generalization of the quantum Yang-Baxter equation [2,3,37,38].…”

Dynamical Yang-Baxter maps and Hopf algebroids associated with s-sets

“…Several studies clarified how to construct the dynamical Yang-Baxter maps [19,20,26,27,29]. In [13], suitable homogeneous pre-systems produce the dynamical Yang-Baxter maps.…”