Integrable Hamiltonian systems on affine Poisson varieties

Abstract

In this chapter we give the basic definitions and properties of integrable Hamiltonian systems on affine Poisson varieties and their morphisms. In Section 2 we define the notion of a Poisson bracket (or Poisson structure) on an affine algebraic variety (over C). The Poisson bracket is precisely what is needed to define Hamiltonian mechanics on a space, as is well-known from the theory of symplectic and Poisson manifolds. We shortly describe the simplest Poisson structures (i.e., linear, affine and quadratic structures; also structures on C² and C³) and describe two natural decompositions of affine Poisson varieties, one is given by the algebra of Casimirs, the other comes from the notion of rank of a Poisson structure (at a point). We also describe several ways to build new affine Poisson varieties from old ones.