Graded identities of simple real graded division algebras

TL;DR

This paper characterizes when two finite dimensional simple real graded division algebras share the same graded identities, linking their structure to matrix algebras over division graded algebras with specific gradings.

Contribution

It provides necessary and sufficient conditions for the coincidence of graded identities and classifies simple real graded algebras based on their graded identities.

Findings

Conditions for coincidence of graded identities of simple real graded division algebras.

Every such algebra shares identities with a matrix algebra over a division graded algebra.

Identifies when two graded algebras have the same graded identities.

Abstract

Let A and B be finite dimensional simple real algebras with division gradings by an abelian group G. In this paper we give necessary and sufficient conditions for the coincidence of the graded identities of A and B. We also prove that every finite dimensional simple real algebra with a G-grading satisfies the same graded identities as a matrix algebra over an algebra D with a division grading that is either a regular grading or a non-regular Pauli grading. Moreover we determine when the graded identities of two such algebras coincide. For graded simple algebras over an algebraically closed field it is known that two algebras satisfy the same graded identities if and only if they are isomorphic as graded algebras.