Varieties of bicommutative algebras

Abstract

Bicommutative algebras are nonassociative algebras satisfying the polynomial identities of right- and left-commutativity (xy)z=(xz)y and x(yz)=y(xz). We study subvarieties of the variety of all bicommutative algebras over a field of characteristic 0. We prove that if the subvariety satisfies a polynomial identity of degree k, then its codimension sequence is bounded by a polynomial of degree k-1. Since the exponent of the variety of all bicommutative algebras is equal to 2, this gives that it is minimal with respect to the codimension growth. Up to isomorphism there are three one-generated two-dimensional bicommutative algebras with one-dimensional square which are nonassociative. We present bases of their polynomial identities and show that one of these algebras generates the whole variety of bicommutative algebras.