ar X iv : q - a lg / 9 60 90 14 v 1 1 6 Se p 19 96 BOUNDARY SOLUTIONS OF THE CLASSICAL YANG - BAXTER EQUATION

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 to 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 sl(n) 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 non-boundary solutions for the classical simple Lie algebras. 0. Introduction. Let g be a simple Lie algebra, either over C or split over R. To fix the terminology call an r ∈ g ∧ g a (constant, “unitary”) solution to the classical Yang-Baxter equation (CYBE) if 〈r, r〉 := [r12, r13] + [r12, r23] + [r13, r23] = 0, and call it a solution to the modified CYBE (MCYBE) if instead 〈r, r〉 is a non-zero multiple of the unique, up to scalar multiple, invariant of ∧3 g. The simply connected algebraic group G with Lie (G) = g operates on both sets, which together with multiplication by non-zero scalars defines a natural concept of equivalence. The solutions to the MCYBE have been constructivly classified by Belavin and Drinfel’d and depend on certain “admissible triples” involving the root system of g, see [BD], or [CP, §3.1.A and §3.1.B]. These triples have the form (Π1,Π2, T ) where Πi is a proper subset of Π, the set of positive simple roots, and T : Π1 → Π2 is a bijection satisfying certain properties (see Definition 2.1 for a precise formulation of these properties). These triples serve as the “discrete” parameter for the space of solutions to the MCYBE; there is also a “continuous” parameter: it is a certain ( d−#Π1 2 ) -dimensional affine subspace of h ∧ h where d is the rank of g and h is a fixed Cartan subalgebra. When g = sl(2) there is only the trivial (or empty)