Classifying G-graded algebras of exponent two

Abstract

Let F be a field of characteristic zero and let V be a variety of associative F-algebras graded by a finite abelian group G. If V satisfies an ordinary non-trivial identity, then the sequence cnG(V) of G-codimensions is exponentially bounded. In [2, 3, 8], the authors captured such exponential growth by proving that the limit G(V)=limn→∞cnG(V)nexists and it is an integer, called the G-exponent of V. The purpose of this paper is to characterize the varieties of G-graded algebras of exponent greater than 2. As a consequence, we find a characterization for the varieties with exponent equal to 2.