Proof of the GGS Conjecture

Abstract

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.