Proof of the Ggs Conjecture

We prove the GGS conjecture [GGS] (1993), which gives a particularly simple explicit quantization of classical r-matrices for Lie algebras gl(n), in terms of a matrix R ∈ Matn(C) ⊗ Matn(C) which satisfies the quantum YangBaxter equation (QYBE) and the Hecke condition, whose quasiclassical limit is r. The r-matrices were classified by Belavin and Drinfeld in the 1980’s in terms of combinatorial objects known as Belavin-Drinfeld triples. We prove this conjecture by showing that the GGS matrix coincides with another quantization from [ESS], which is a more general construction. We do this by explicitly expanding the product from [ESS] using detailed combinatorial analysis in terms of Belavin-Drinfeld triples.