Lifting partial actions: from groups to groupoids

TL;DR

This paper explores the relationship between partial actions of connected groupoids and their isotropy groups, establishing isomorphisms with partial skew group rings and developing related Morita and Galois theories.

Contribution

It introduces a framework connecting partial groupoid actions to partial group actions, including isomorphisms, Morita theory, and properties of associated rings.

Findings

Existence of a datum linking partial groupoid and group actions

Isomorphism between partial skew groupoid rings and partial skew group rings under certain conditions

Development of Morita and Galois theories for these partial actions

Abstract

In this paper, we are interested in the study of the existence of connections between partial groupoid actions and partial group actions. Precisely, we prove that there exists a datum connecting a partial action of a connected groupoid and a partial action of any of its isotropy groups. Furthermore, it will be proved that under a suitable condition the partial skew groupoid ring corresponding to a partial action by a connected groupoid is isomorphic to a specific partial skew group ring. We also present a Morita theory and a Galois theory related to these partial actions as well as considerations about the strictness of the corresponding Morita contexts. Semisimplicity, separability and Frobenius properties of the corresponding partial skew groupoid rings are also considered.