Crossed products by twisted partial actions and graded algebras

TL;DR

This paper studies the structure of crossed products by twisted partial group actions on algebras, providing associativity results and criteria for when a graded algebra is isomorphic to such a crossed product, including stability conditions.

Contribution

It establishes associativity of crossed products by twisted partial actions and provides criteria for graded algebras to be isomorphic to these crossed products, including stability conditions.

Findings

Crossed products by twisted partial actions are associative.

Criteria for graded algebras to be isomorphic to crossed products are established.

Under certain conditions, graded algebras are stably isomorphic to crossed products.

Abstract

For a twisted partial action \Theta of a group G on an (associative non-necessarily unital) algebra A over a commutative unital ring k, the crossed product A X_\Theta G is proved to be associative. Given a G-graded k-algebra B = \oplus_{g\in G}\B_g with the mild restriction of homogeneous non-degeneracy, a criteria is established for B to be isomorphic to the crossed product B_1 X_\Theta G for some twisted partial action of G on B_1. The equality B_g\B_{g^{-1}}B_g = \B_g for all g\in G is one of the ingredients of the criteria, and if it holds and, moreover, B has enough local units, then it is shown that B is stably isomorphic to a crossed product by a twisted partial action of G.