Identities and isomorphisms of finite-dimensional graded simple algebras

TL;DR

This paper proves that two finite-dimensional G-graded simple algebras over an algebraically closed field are isomorphic if and only if they satisfy the same graded polynomial identities, establishing a key classification criterion.

Contribution

It establishes a complete characterization of isomorphism classes of finite-dimensional G-graded simple algebras via their graded polynomial identities.

Findings

A and B are isomorphic iff they satisfy the same graded polynomial identities.

Provides a criterion for classifying finite-dimensional G-graded simple algebras.

Enhances understanding of graded algebra structures and their identities.

Abstract

Let $\mathbb F$ be an algebraically closed field, $G$ be an abelian group, and let $A$ and $B$ be arbitrary finite-dimensional $G$-graded simple algebras over $\mathbb F$. We prove that $A$ and $B$ are isomorphic if, and only if, they satisfy the same graded polynomial identities.