Fundamental theorem of \((A,\mathcal G,H)\)-comodules

Abstract

Let \(k\) be a field, \(H\) a Hopf algebra with a bijective antipode, \(\mathcal G\) an \(H\)-comodule Lie algebra and \(A\) a commutative \(({\mathcal G},H)\)-comodule algebra. We assume that there is an \(H\)-colinear algebra map from \(H\) to \(A^{\mathcal G}\). We generalize the Fundamental Theorem of \((A,H)\)-Hopf modules to \((A,{\mathcal G},H)\)-comodules, and we deduce relative projectivity in the category of \((A,{\mathcal G},H)\)-comodules. In many applications, \(A\) could be a commutative \(G\)-graded \(\mathcal G\)-module algebra, where \(G\) is an abelian group and \(\mathcal G\) is a \(G\)-graded Lie algebra;  or a rational \(({\mathcal G},G)\)-module algebra, where \(G\) is an affine algebraic group and \(\mathcal G\) is a rational \(G\)-module Lie algebra.