Trigonometric solutions of the associative Yang-Baxter equation

TL;DR

This paper classifies trigonometric solutions to the associative Yang-Baxter equation for matrix algebras, revealing their relation to quantum Yang-Baxter solutions and classical r-matrices, and providing explicit existence conditions.

Contribution

It establishes a connection between AYBE solutions and quantum Yang-Baxter solutions, with explicit criteria based on Belavin-Drinfeld triples.

Findings

AYBE solutions correspond to scaled quantum Yang-Baxter solutions

Existence of associative lifts depends on classical r-matrix classification

Results link AYBE with classical and quantum bialgebras

Abstract

We classify trigonometric solutions to the associative Yang-Baxter equation (AYBE) for A = Mat_n, the associative algebra of n-by-n matrices. The AYBE was first presented in a 2000 article by Marcelo Aguiar and also independently by Alexandre Polishchuk. Trigonometric AYBE solutions limit to solutions of the classical Yang-Baxter equation. We find that such solutions of the AYBE are equal to special solutions of the quantum Yang-Baxter equation (QYBE) classified by Gerstenhaber, Giaquinto, and Schack (GGS), divided by a factor of q - q^{-1}, where q is the deformation parameter q = exp(h). In other words, when it exists, the associative lift of the classical r-matrix coincides with the quantum lift up to a factor. We give explicit conditions under which the associative lift exists, in terms of the combinatorial classification of classical r-matrices through Belavin-Drinfeld triples. The results of this paper illustrate nontrivial connections between the AYBE and both classical (Lie) and quantum bialgebras.