R-Brackets and r-Brackets

Abstract

An R-matrix or an r-matrix is an additional structure on a Lie algebra, which yields a new Lie bracket on the Lie algebra or on its dual, hence leading to a Poisson structure on the dual Lie algebra or on the Lie algebra itself (in that order). Both constructions are discussed, as well as their relation when $\mathfrak{g}$ is a quadratic Lie algebra, i.e., when $\mathfrak{g}$ is equipped with a non-degenerate symmetric bilinear form, which is ad-invariant. We also show how R or r-matrices lead in the case of an associative Lie algebra to quadratic and cubic Poisson structures and how all these Poisson brackets are related via Lie derivatives. The connection with Poisson–Lie groups is discussed in the next chapter.