On Pre-Novikov Algebras and Derived Zinbiel Variety

TL;DR

This paper explores derived varieties of non-associative algebras with derivations, establishing conditions for embeddability into differential algebras and analyzing specific cases like Zinbiel algebras and their derived pre-Novikov structures.

Contribution

It provides a sufficient condition for embedding derived algebras into differential algebras and examines the structure of derived varieties, including non-embeddable examples.

Findings

Derived varieties can often be embedded into differential algebras under certain conditions.

The derived variety of Zinbiel algebras coincides with pre-Novikov algebras.

Some derived algebras cannot be embedded into Zinbiel algebras with derivations.

Abstract

For a non-associative algebra $A$ with a derivation $d$, its derived algebra $A^{(d)}$ is the same space equipped with new operations $a\succ b = d(a)b$, $a\prec b = ad(b)$, $a,b\in A$. Given a variety ${\rm Var}$ of algebras, its derived variety is generated by all derived algebras $A^{(d)}$ for all $A$ in ${\rm Var}$ and for all derivations $d$ of $A$. The same terminology is applied to binary operads governing varieties of non-associative algebras. For example, the operad of Novikov algebras is the derived one for the operad of (associative) commutative algebras. We state a sufficient condition for every algebra from a derived variety to be embeddable into an appropriate differential algebra of the corresponding variety. We also find that for ${\rm Var} = {\rm Zinb}$, the variety of Zinbiel algebras, there exist algebras from the derived variety (which coincides with the class of pre-Novikov algebras) that cannot be embedded into a Zinbiel algebra with a derivation.