On sums of gr-PI algebras

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 polynomials 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 Kȩpczyk 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 group graded Lie algebras. We note that there is no known analogue of Kȩpczyk's theorem for Lie algebras.