Identities of finitely generated graded algebras with involution

TL;DR

This paper investigates the identities of finitely generated graded algebras with involution, establishing conditions for PI-representability and extending classical theorems to the graded involution setting.

Contribution

It proves that finitely generated graded PI-algebras with involution have the same identities as some finite dimensional graded algebra with involution, extending Kemer's theorem.

Findings

Finitely generated graded PI-algebras with involution are PI-representable by finite dimensional algebras.

The identities of such algebras coincide with those of finite dimensional counterparts for prime q or q=4.

Results extend classical PI-theory to graded algebras with involution.

Abstract

We consider associative algebras with involution graded by a finite abelian group G over a field of characteristic zero. Suppose that the involution is compatible with the grading. We represent conditions permitting PI-representability of such algebras. Particularly, it is proved that a finitely generated (Z/qZ)-graded associative PI-algebra with involution satisfies exactly the same graded identities with involution as some finite dimensional (Z/qZ)-graded algebra with involution for any prime q or q = 4. This is an analogue of the theorem of A.Kemer for ordinary identities, and an extension of the result of the author for identities with involution. The similar results were proved also recentely for graded identities.