On sums of gr-PI algebras

TL;DR

This paper investigates conditions under which the sum of two graded subalgebras inherits polynomial identities, revealing limitations of existing theorems and introducing new concepts like graded semi-identities.

Contribution

It proves that graded identities are preserved under certain conditions when summing subalgebras, introduces graded semi-identities, and provides counterexamples to previous theorems.

Findings

If B is an ideal and both B and C satisfy graded identities, then A=B+C also satisfies them.

Counterexample showing B and C can satisfy identities while A does not.

Extension of concepts to graded Lie algebras and discussion of limitations of existing theorems.

Abstract

Let $A=B+C$ be an associative algebra graded by a group $G$, which is a sum of two homogeneous subalgebras $B$ and $C$. We prove that if $B$ is an ideal of $A$, and both $B$ and $C$ satisfy graded polynomial identities, then the same happens for the algebra $A$. We also introduce the notion of graded semi-identity for the algebra $A$ graded by a finite group and we give sufficient conditions on such semi-identities in order to obtain the existence of graded identities on $A$. We also provide an example where both subalgebras $B$ and $C$ satisfy graded identities while $A=B+C$ does not. Thus the theorem proved by K\c{e}pczyk in 2016 does not transfer to the case of group graded associative algebras. A variation of our example shows that a similar statement holds in the case of graded group Lie algebras. We note that there is no known analogue of K\c{e}pczyk's theorem for Lie algebras.