Commutative non-power-associative algebras

Abstract

We study commutative algebras satisfying the identity [Formula: see text] It is known that for [Formula: see text] and for characteristic not [Formula: see text] or [Formula: see text], the algebra is a commutative power-associative algebra. These algebras have been widely studied by Albert, Gerstenhaber and Schafer. For [Formula: see text] Guzzo and Behn in 2014 proved that commutative algebras of dimension [Formula: see text] satisfying [Formula: see text] are solvable. We consider the remaining values of [Formula: see text] We prove that commutative algebras satisfying [Formula: see text] with [Formula: see text] and generated by one element are nilpotent of nilindex [Formula: see text] (we assume characteristic of the field [Formula: see text]).