1999/03/13 by Travis Schedler, Schedler, Travis
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #math.QA
paper · pdf · doi:10.48550/arxiv.math/9903079
24 pages, AMSLaTeX. Includes an appendix by Pavel Etingof and the author. Completely rewritten from version 1, with many new results added
openalex publication_date 1999/03/13 · arxiv created 1999/08/11 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ≠ r + r21 ∈ (S2 \mathfrakg)^\mathfrakg. 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 \mathfraksl(n). As a result, they formulated a conjecture stating that certain explicitly given elements R ∈ Matn(\mathbb C) ⊗ Matn(\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 An-1. Following a suggestion from Gerstenhaber and Giaquinto, we propose an alternate form for R, given by RJ = qr0 J-1 Rs J21 qr0, for a suitable twist J and a diagonal matrix r0, where Rs is the standard Drinfeld-Jimbo solution of the QYBE. We formulate the ``twist conjecture'', which states that RJ = RGGS and that RJ satisfies the QYBE. Since RJ by construction satisfies the Hecke relation, this conjecture implies the GGS conjecture. We check the twist conjecture by computer for n ≤ 12 and show that it is true modulo ℏ3. We provide combinatorial formulas for coefficients in the matrices RJ, RGGS and prove both conjectures in the disjoint case---when Γ1 ∩ Γ2 = ∅---and in the orthogonal generalized disjoint case, which is a generalization of Γ1 ⊥ Γ2. Finally, we prove the twist conjecture for the Cremmer-Gervais triple and discuss cases in which it is known that RJ = RGGS.