A correspondence between homogeneous and Galois coactions of Hopf algebras

TL;DR

This paper establishes a duality between homogeneous and Galois coactions of Hopf algebras, linking their structures through Morita equivalence and isomorphism, advancing the understanding of symmetries in algebraic systems.

Contribution

It introduces a duality framework connecting homogeneous and Galois coactions of Hopf algebras, up to Morita equivalence and isomorphism.

Findings

Duality between homogeneous and Galois coactions established

Morita equivalence classifies homogeneous coactions

Isomorphism classifies Galois coactions

Abstract

A coaction of a Hopf algebra on a unital algebra is called homogeneous if the algebra of coinvariants equals the ground field. A coaction of a Hopf algebra on a (not necessarily unital) 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 coactions, up to equivariant Morita equivalence, and Galois coactions, up to isomorphism.