On the GGS Conjecture

Abstract

In the 1980's, Belavin and Drinfeld classified solutions r of the classical Yang-Baxter equation (CYBE) for simple Lie algebras \mathfrak g satisfying 0 \neq r + r_{21} \in (S^2 \mathfrak{g})^{\mathfrak{g}}. They proved that all such solutions fall into finitely many continuous families and introduced combinatorial objects to label these families, Belavin-Drinfeld triples. In 1993, Gerstenhaber, Giaquinto, and Schack attempted to quantize such solutions for Lie algebras \mathfrak{sl}(n). As a result, they formulated a conjecture stating that certain explicitly given elements R \in Mat_n(\mathbb C) \otimes Mat_n(\mathbb C) satisfy the quantum Yang-Baxter equation (QYBE) and the Hecke relation. Specifically, the conjecture assigns a family of such elements R to any Belavin-Drinfeld triple of type A_{n-1}. Following a suggestion from Gerstenhaber and Giaquinto, we propose an alternate form for R, given by R_J = q^{r^0} J^{-1} R_s J_{21} q^{r^0}, for a suitable twist J and a diagonal matrix r^0, where R_s is the standard Drinfeld-Jimbo solution of the QYBE. We formulate the ``twist conjecture'', which states that R_J = R_{\text{GGS}} and that R_J satisfies the QYBE. Since R_J by construction satisfies the Hecke relation, this conjecture implies the GGS conjecture. We check the twist conjecture by computer for n \leq 12 and show that it is true modulo \hbar^3. We provide combinatorial formulas for coefficients in the matrices R_J, R_{\text{GGS}} and prove both conjectures in the disjoint case---when Γ_1 \cap Γ_2 = \emptyset---and in the orthogonal generalized disjoint case, which is a generalization of Γ_1 \perp Γ_2. Finally, we prove the twist conjecture for the Cremmer-Gervais triple and discuss cases in which it is known that R_J = R_{\text{GGS}}.