r-Matrices and Yang Baxter Equations

Abstract

One way to prove involutivity of a set of conserved quantities Q _k = TrL ^k, k = 1, 2, ..., for a Lax equation $\dot L$ = [L, M], with L and M lying in some Lie algebra $\mathcal{G}$, is to find an element r in the tensor product $\mathcal{G} \otimes \mathcal{G}$ satisfying

(7.1)

For simplicity, we will assume that $\mathcal{G}$ is finite dimensional and is realized as a Lie algebra of matrices acting on a finite dimensional (real or complex) vectorspace V with fixed orthonormal basis {e _i }, i = 1, ..., N. So L and M will simply be (real or complex) (N × N matrices.