Verification of the GGS conjecture for sl(n), n <= 12

Abstract

In the 1980's, Belavin and Drinfeld classified non-unitary solutions of the classical Yang-Baxter equation (CYBE) for simple Lie algebras. 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 sl(n). As a result, they formulated a conjecture stating that certain explicitly given elements R satisfy the quantum Yang-Baxter equation (QYBE) and the Hecke condition. Specifically, the conjecture assigns a family of such elements R to any Belavin-Drinfeld triple of type A_{n-1}. Until recently, this conjecture has only been known to hold for n <= 4. In 1998 Giaquinto and Hodges checked the conjecture for n=5 by direct computation using Mathematica. Here we report a computation which allowed us to check that the conjecture holds for n <= 10. The program is included which prints an element R for any triple and checks that R satisfies the QYBE and Hecke conditions.