Linear Poisson Structures and Lie Algebras

Abstract

Together with symplectic manifolds, considered in the previous chapter, Lie algebras provide the first examples of Poisson manifolds. The dual $\mathfrak{g}^{*}$ of a finite-dimensional Lie algebra $\mathfrak{g}$ admits a natural Poisson structure, called its Lie–Poisson structure. It is a linear Poisson structure and every linear Poisson structure (on a finite-dimensional vector space) is a Lie–Poisson structure. We show that the leaves of the symplectic foliation are the coadjoint orbits of the adjoint group of $\mathfrak{g}$ and we shortly discuss the linearization of Poisson structures (in the neighborhood of a point where the rank is zero). Using a non-degenerate Ad-invariant symmetric bilinear form, we get the Lie-Poisson structure on $\mathfrak{g}$, which has several virtues, amongst which the fact that the Hamiltonian vector fields on $\mathfrak{g}$ take a natural form, a so-called Lax form. Affine Poisson structures and their Lie theoretical interpretation are discussed at the end of the chapter.