Strong equivalence of graded algebras

TL;DR

This paper introduces a new notion of strong equivalence for graded algebras, demonstrating how various gradings relate to skew group algebras and Morita equivalence, with implications for algebra classification.

Contribution

It defines strong equivalence of graded algebras, links partial and global actions, and shows preservation of gradings and Morita equivalence under these relations.

Findings

Partially-strongly-graded algebras are strongly-graded-equivalent to skew group algebras.

Cohen-Montgomery duality extends to certain idempotent graded algebras.

Strongly-graded-equivalence preserves strong gradings and relates to Morita equivalence.

Abstract

We introduce the notion of a strong equivalence between graded algebras and prove that any partially-strongly-graded algebra by a group $G$ is strongly-graded-equivalent to the skew group algebra by a product partial action of $G$. As to a more general idempotent graded algebra $B$, we point out that the Cohen-Montgomery duality holds for $B$, and $B$ is graded-equivalent to a global skew group algebra. We show that strongly-graded-equivalence preserves strong gradings and is nicely related to Morita equivalence of product partial actions. Furthermore, we prove that any product partial group action $\alpha $ is globalizable up to Morita equivalence; if such a globalization $\beta $ is minimal, then the skew group algebras by $\alpha $ and $\beta $ are graded-equivalent; moreover, $\beta $ is unique up to Morita equivalence. Finally, we show that strongly-graded-equivalent partially-strongly-graded algebras are stably isomorphic as graded algebras.