Comodule algebras and galois extensions relative to polynomial algebras, free algebras, and enveloping algebras

Abstract

In this paper we study the question of when an H -comodule algebra is a faithfully flat Galois extension of its sub algebra of coinvariants for certain Hopf algebras H. We note that if H is connected, a faithfully flat Galois extension must actually be cleft and hence a crossed product, and we show that with a different hypothesis, a faithfully flat Galois extension must be a smash product. We also describe faithfully flat Galois extensions when H is pointed cocommutative. We give an explicit description of H -comodule algebras when H is a polynomial algebra, a divided power Hopf algebra, a free algebra, or a shuffle algebra. We give necessary and sufficient conditions for an H -extension to be faithfully flat Galois in these cases and in the case where H is the enveloping algebra of a Lie algebra; a key ingredient in our analysis is the existence and description of a total integral. In the case where , we give a simple example of a flat Galois extension that is not faithfully flat.