Polynomial identities in Novikov algebras

TL;DR

This paper investigates polynomial identities in Novikov algebras over fields of zero characteristic, revealing universal identities and their implications for algebra structure, including Lie-solvability and finite basis properties.

Contribution

It establishes that Novikov algebras satisfying nontrivial identities must satisfy universal identities, and proves finite basis results for identities over zero characteristic fields.

Findings

Novikov algebras satisfying identities are Lie-solvable

Universal identities like right associator nilpotence hold in such algebras

Systems of identities are finitely based over zero characteristic fields

Abstract

In this paper, we study Novikov algebras satisfying nontrivial identities. We show that a Novikov algebra over a field of zero characteristic that satisfies a nontrivial identity satisfies some unexpected "universal" identities, in particular, right associator nilpotence, and right nilpotence of the commutator ideal. This, in particular, implies that a Novikov algebra over a field of zero characteristic satisfies a nontrivial identity if and only if it is Lie-solvable. We also establish that any system of identities of Novikov algebras over a field of zero characteristic follows from finitely many of them, and that the same holds over any field for multilinear Novikov identities. Some analogous simpler statements are also proved for commutative differential algebras.