Subprime solutions of the classical Yang-Baxter equation

We introduce a new family of classical r-matrices for the Lie algebra sl(n) 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 phi(n) of these components are SLn-orbits of single points. These points are the generalized Cremmer-Gervais r-matrices r(i,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 p(i,n) subset of sl(n) as carriers. We prove this conjecture in the cases when n equivalent to +/- 1 (mod i). The subprime linear functionals f is an element of p(i,n)* and the corresponding principal elements H is an element of p(i,n) play important roles in our proof. Since the subprime functionals are Frobenius precisely in the cases when n equivalent to +/- 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. (C) 2018 Elsevier Inc. All rights reserved.