  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.
