2017/12/19 by Garrett Johnson, Johnson, Garrett
Mathematics · #16T25 #17B62 #FOS: Mathematics #Quantum Algebra (math.QA) #math.QA #msc:16T25 #msc:17B62
paper · pdf · doi:10.48550/arxiv.1712.07258
16 pages, v2 includes proofs of Lemmas 4.2 and 4.3, to appear in Journal of Algebra
arxiv created 2018/09/26 · arxiv updated 2018/09/28
We introduce a new family of classical r-matrices for the Lie algebra \mathfraksln 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 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 \mathfrakpi,n⊆ \mathfraksln as carriers. We prove this conjecture in the cases when n≡ ± 1 (mod i). The subprime linear functionals f∈\mathfrakpi, n^* and the corresponding principal elements H∈\mathfrakpi, 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.