Boundary Solutions of the Classical Yang--Baxter Equation

Abstract

We define a new class of unitary solutions to the classical Yang--Baxter equation (CYBE). These ‘boundary solutions’ are those which lie in the closure of the space of unitary solutions of the modified classical Yang--Baxter equation (MCYBE). Using the Belavin--Drinfel'd classification of the solutions to the MCYBE, we are able to exhibit new families of solutions to the CYBE. In particular, using the Cremmer--Gervais solution to the MCYBE, we explicitly construct for all n ≥ 3 a boundary solution based on the maximal parabolic subalgebra of $${\mathfrak{s}}{\mathfrak{l}}\left( n \right)$$ obtained by deleting the first negative root. We give some evidence for a generalization of this result pertaining to other maximal parabolic subalgebras whose omitted root is relatively prime to n. We also give examples of nonboundary solutions for the classical simple Lie algebras.