Graded polynomial identities as identities of universal algebras

TL;DR

This paper proves that finite-dimensional simple graded algebras over an algebraically closed field are uniquely determined by their graded identities, leading to isomorphism when these identities coincide.

Contribution

It establishes that identical graded identities imply isomorphism for finite-dimensional simple graded algebras over algebraically closed fields.

Findings

Graded identities characterize simple graded algebras uniquely.

If two such algebras share the same graded identities, they are isomorphic.

The result applies to algebras graded by any semigroup over an algebraically closed field.

Abstract

Let $A$ and $B$ be finite-dimensional simple algebras with arbitrary signature over an algebraically closed field. Suppose $A$ and $B$ are graded by a semigroup $S$ so that the graded identitical relations of $A$ are the same as those of $B$. Then $A$ is isomorphic to $B$ as an $S$-graded algebra.