1 Verification of the GGS conjecture for sl(n), n ≤ 10.

In the 1980’s, Belavin and Drinfeld classified non-unitary solutions of the classical Yang-Baxter equation (CYBE) for simple Lie algebras [1]. 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 ∈ Matn(C)⊗Matn(C) satisfy the quantum Yang-Baxter equation (QYBE) and the Hecke condition [2]. Specifically, the conjecture assigns a family of such elements R to any Belavin-Drinfeld triple of type An−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 [3]. 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. 1 Belavin-Drinfeld triples Let (ei), 1 ≤ i ≤ n, be a basis for C . Set Γ = {ei − ei+1 : 1 ≤ i ≤ n − 1}. We will use the notation αi ≡ ei − ei+1. Let (, ) denote the inner product on C n having (ei) as an orthonormal basis. Definition 1.1 A Belavin-Drinfeld triple of type An−1 is a triple (τ,Γ1,Γ2) where Γ1,Γ2 ⊂ Γ and τ : Γ1 → Γ2 is a bijection, satisfying two conditions: (a) ∀α, β ∈ Γ1, (τα, τβ) = (α, β). (b) τ is nilpotent: ∀α ∈ Γ1, ∃k ∈ N such that τ α / ∈ Γ1. We employ three isomorphisms of Belavin-Drinfeld triples: a) Any triple (τ,Γ1,Γ2) is isomorphic to the triple (τ ,Γ1,Γ ′ 2) obtained as follows: Γ ′ 1 = {αm : αn−m ∈ Γ1}, τ (αm) = αk where τ(αn−m) = αn−k. b) Any triple (τ,Γ1,Γ2) is isomorphic to the triple (τ ,Γ2,Γ1). c) The product of isomorphisms (a), (b). Modulo these isomorphisms, we found all Belavin-Drinfeld triples for n ≤ 10 by computer. The number of such triples is given below: n # of triples n # of triples n # of triples 2 1 5 13 8 611 3 2 6 41 9 2490 4 4 7 161 1