A Characterization of Varieties of Associative Algebras of Exponent two

Abstract

It was recently proved that any variety of associative algebras over a field of characteristic zero has an integral exponential growth. It is known that a variety V has polynomial growth if and only if V does not contain the Grassmann algebra and the algebra of 2 × 2 upper triangular matrices. It follows that any variety with overpolynomial growth has exponent at least 2. In this note we characterize varieties of exponent 2 by exhibiting a finite list of algebras playing a role similar to the one played by the two algebras above.