Distributive lattices of varieties of Novikov algebras

TL;DR

This paper characterizes when the lattice of subvarieties of Novikov algebras is distributive, linking it to degree three identities, and classifies related Koszul operads, answering a long-standing question.

Contribution

It establishes a criterion for distributivity of subvariety lattices in Novikov algebras and classifies certain Koszul operads related to the Novikov operad.

Findings

Distributivity of subvariety lattice is equivalent to degree three identities.

Classified all Koszul operads with one binary generator related to Novikov operad.

Answered a fifty-year-old question of Bokut.

Abstract

We prove that a variety of Novikov algebras has a distributive lattice of subvarieties if and only if the lattice of its subvarieties defined by identities of degree three is distributive, thus answering, in the case of Novikov algebras, a question of Bokut from about fifty years ago. As a byproduct, we classify all Koszul operads with one binary generator of which the Novikov operad is a quotient.