Partial Hopf actions, partial invariants and a Morita context

TL;DR

This paper investigates properties of partial Hopf actions, focusing on fixed point subalgebras, Morita contexts, and Hopf-Galois extensions, extending classical results to the partial action setting.

Contribution

It constructs a Morita context for partial actions of finite dimensional Hopf algebras, generalizing known results and exploring fixed point subalgebras and Galois extensions.

Findings

Existence of enveloping actions for partial Hopf actions

Construction of Morita context relating fixed point subalgebra and partial smash product

Reinterpretation of classical Hopf-Galois results in the partial action context

Abstract

Partial actions of Hopf algebras can be considered as a generalization of partial actions of groups on algebras. Among important properties of partial Hopf actions, it is possible to assure the existence of enveloping actions. This allows to extend several results from the theory of partial group actions to the Hopf algebraic setting. In this article, we explore some properties of the fixed point subalgebra with relations to a partial action of a Hopf algebra. We also construct, for partial actions of finite dimensional Hopf algebras a Morita context relating the fixed point subalgebra and the partial smash product. This is a generalization of a well known result in the theory of Hopf algebras for the case of partial actions. Finally, we study Hopf-Galois extensions and reobtain some classical results in the partial case.