Reduction

Abstract

We explain in this chapter a few more advanced constructions, which are related to the general concept of reduction. Roughly speaking, reduction means that the object under study (here a Poisson structure), is replaced by an object of the same type, but on a manifold of smaller dimension, in classical terms “with less degrees of freedom”. Poisson reduction deals with Poisson structures on quotients of Poisson manifolds or of coisotropic submanifolds of Poisson manifolds. Poisson–Dirac reduction is concerned with Poisson structures on submanifolds, which are not necessarily Poisson submanifolds. As examples and applications, we consider the transverse Poisson structure to an arbitrary symplectic leaf of a Poisson manifold, fixed point sets and quotient spaces of groups acting on Poisson varieties/manifolds and the reduced space of the momentum map, associated to the action. We also describe each of the constructions in the corresponding algebraic context.