Trigonometric solutions of the associative Yang-Baxter equation

We classify trigonometric solutions to the associative Yang-Baxter equation (AYBE) for $A = \mathrm{Mat}_n$, 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 $q = e^{\hbar}$. 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.

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Published 1 January 2003