Trigonometric solutions of the associative Yang-Baxter equation

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.Comment: 20 pages, AMSLaTeX with BibTeX references and the MRL article class. v2 includes minor correction

Religion and Hip Hop

Book review of Religion and Hip Hop, by Monica Miller (2013)

Proof of the GGS Conjecture

We prove the GGS conjecture (1993), due to Gerstenhaber, Giaquinto, and Schack, which gives a particularly simple explicit quantization of classical r-matrices for Lie algebras gl(n) in terms of an element R satisfying the quantum Yang-Baxter equation and the Hecke condition. The r-matrices were classified by Belavin and Drinfeld in the 1980s in terms of combinatorial objects known as Belavin-Drinfeld triples. We prove this conjecture by showing that the GGS matrix coincides with another quantization due to Etingof, Schiffmann, and the author, which is a more general construction. We do this by explicitly expanding the product from the aforementioned paper using detailed combinatorial analysis in terms of Belavin-Drinfeld triples.Comment: AMSLaTeX; uses mrlart2e.cls (included-- MRL's document class, based on amsart