Subprime Solutions of the Classical Yang-Baxter Equation

We introduce a new family of classical $r$-matrices for the Lie algebra $\mathfrak{sl}_n$ that lies in the Zariski boundary of the Belavin-Drinfeld space ${\mathcal M}$ of quasi-triangular solutions to the classical Yang-Baxter equation. In this setting ${\mathcal M}$ is a finite disjoint union of components; exactly $\phi(n)$ of these components are $SL_n$-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 $\mathfrak{p}_{i,n}\subseteq \mathfrak{sl}_n$ as carriers. We prove this conjecture in the cases when $n\equiv \pm 1$ (mod $i$). The subprime linear functionals $f\in\mathfrak{p}_{i, n}^*$ and the corresponding principal elements $H\in\mathfrak{p}_{i, n}$ play important roles in our proof. Since the subprime functionals are Frobenius precisely in the cases when $n\equiv \pm 1$ (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.