Subprime solutions of the classical Yang–Baxter equation

Abstract

We introduce a new family of classical r-matrices for the Lie algebra sln that lies in the Zariski boundary of the Belavin–Drinfeld space M of quasi-triangular solutions to the classical Yang–Baxter equation. In this setting M is a finite disjoint union of components; exactly ϕ(n) of these components are SLn-orbits of single points. These points are the generalized Cremmer–Gervais r-matrices ri,n which are naturally indexed by pairs of positive coprime integers, i and n, with i<n. A conjecture of Gerstenhaber and Giaquinto states that the boundaries of the Cremmer–Gervais components contain r-matrices having maximal parabolic subalgebras pi,n⊆sln as carriers. We prove this conjecture in the cases when n≡±1 (mod i). The subprime linear functionals f∈pi,n⁎ and the corresponding principal elements H∈pi,n play important roles in our proof. Since the subprime functionals are Frobenius precisely in the cases when n≡±1 (mod i), this partly explains our need to require these conditions on i and n. We conclude with a proof of the GG boundary conjecture in an unrelated case, namely when (i,n)=(5,12), where the subprime functional is no longer a Frobenius functional.