Multiplicative and Quadratic Poisson Structures

Abstract

After studying linearization and normal forms for Poisson structures with a non-trivial linear part in the previous two chapters, it is a logical next step to talk about Poisson structures which begin with a quadratic term. This leads us to the study of (homogeneous) quadratic Poisson structures, and then to another important subject of Poisson geometry, namely Poisson—Lie groups. The reason is that both Poisson—Lie groups and quadratic Poisson structures are in a sense multiplicative, and they both arise, most of the times, from classical r-matrices. For example, if we have a multiplicative Poisson structure on a finite-dimensional unital associative algebra (say the algebra Mat _n (ℝ) of nxn-matrices), then it is a quadratic Poisson structure, and at the same time a Poisson—Lie structure on the group of reversible elements of the algebra. In this chapter, we will give a brief introduction to the theory of Poisson—Lie groups, and then discuss some results on quadratic Poisson structures and the quadratization problem.