Fundamental Theorem of Poisson (A,H)-Hopf modules

Abstract

Let k be a field, H a Hopf algebra with a bijective antipode and A an H-comodule Poisson algebra. We assume that there is an H-colinear algebra map from H to the Poisson-center of A. We generalize the Fundamental Theorem of (A,H)-Hopf modules to Poisson (A,H)-Hopf modules, and we deduce relative projectivity in the category of Poisson (A,H)-Hopf modules. In many applications, A could be a rational G-module Poisson algebra, where G is an affine algebraic group or the coordinate ring of an affine Poisson variety over C on which acts morphically an algebraic group via automorphisms of Poisson varieties.