Graded Embeddings of Finite Dimensional Simple Graded Algebras

TL;DR

This paper investigates conditions under which finite dimensional G-graded algebras can be embedded into each other based on their graded identities, providing a characterization for G-simple and G-semisimple cases.

Contribution

It establishes a criterion linking graded embeddings to the inclusion of graded identities for G-simple and G-semisimple algebras over an algebraically closed field.

Findings

Embedding of G-simple algebras corresponds to graded identity inclusion.

Weaker conditions are provided for G-semisimple algebras.

Results apply to finite dimensional algebras over characteristic zero fields.

Abstract

Let A,B be finite dimensional G-graded algebras over an algebraically closed field K with char(K)=0, where G is an abelian group, and let Id_G(A) be the set of graded identities of A (res. Id_G(B)). We show that if A,B are G-simple then there is a graded embedding of A in B iff Id_G(B) is contained in Id_G(A). We also give a weaker generalization for the case where A is G-semisimple and B is arbitrary.