On Pre-Novikov Algebras and Derived Zinbiel Variety

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 $\mathrm{Var}$ of algebras, its derived variety is generated by all derived algebras $A^{(d)}$ for all $A$ in $\mathrm{Var}$ 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 $\mathrm{Var}=\mathrm{Zinb}$, 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.

References in corpus (2)

• Noncommutative Novikov algebras
• One-dimensional central extensions and simplicities of a class of left-symmetric conformal algebras