Duality for partial group actions

TL;DR

This paper extends Cohen-Montgomery duality to partial group actions, showing isomorphisms involving partial skew group rings and matrix algebras, and explores duality in infinite and Hopf action contexts.

Contribution

It generalizes duality results from global to partial group actions, establishing structural isomorphisms and separability properties for partial skew group rings.

Findings

Partial skew group rings are isomorphic to direct products involving matrix algebras.

The partial skew group ring embeds as a separable subalgebra of a matrix algebra.

Duality concepts extend to infinite partial actions and partial Hopf actions.

Abstract

Given a finite group G acting as automorphisms on a ring A, the skew group ring A*G is an important tool for studying the structure of G-stable ideals of A. The ring A*G is G-graded, i.e.G coacts on A*G. The Cohen-Montgomery duality says that the smash product A*G#k[G]^* of A*G with the dual group ring k[G]^* is isomorphic to the full matrix ring M_n(A) over A, where n is the order of G. In this note we show how much of the Cohen-Montgomery duality carries over to partial group actions in the sense of R.Exel. In particular we show that the smash product (A*_\alpha G)#k[G]^* of the partial skew group ring A*_\alpha G and k[G]^* is isomorphic to a direct product of the form K x eM_n(A)e where e is a certain idempotent of M_n(A) and K is a subalgebra of (A *_\alpha G)#k[G]^*. Moreover A*_\alpha G is shown to be isomorphic to a separable subalgebra of eM_n(A)e. We also look at duality for infinite partial group actions and for partial Hopf actions.