Codimension growth of two-dimensional non-associative algebras

Abstract

Let $F$ be a field of characteristic zero and let $A$ be a two- dimensional non-associative algebra over $F$. We prove that the sequence $c_n(A), n\!=\!1,2,\ldots ,$ of codimensions of $A$ is either bounded by $n+1$ or grows exponentially as $2^n$. We also construct a family of two-dimensional algebras indexed by rational numbers with distinct T-ideals of polynomial identities and whose codimension sequence is $n+1$, $n\ge 2$.