A Correspondence Between Homogeneous and Galois Coactions of Hopf Algebras

Let H be a Hopf algebra. A unital H-comodule algebra is called homogeneous if the algebra of coinvariants equals the ground field. A (not necessarily unital) H-comodule algebra is called Galois, or principal, or free, if the canonical map, also known as the Galois map, is bijective. In this paper, we establish a duality between a particular class of homogeneous H-comodule algebras, up to H-Morita equivalence, and a particular class of Galois H-comodule algebras, up to H-comodule algebra isomorphism.