On Pre-Novikov Algebras and Derived Zinbiel Variety

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 $\mathrm{V}\mathrm{a}\mathrm{r}$ of algebras, its derived variety is generated by all derived algebras $A^{(d)}$ for all $A$ in $\mathrm{V}\mathrm{a}\mathrm{r}$ 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 $\mathrm{V}\mathrm{a}\mathrm{r}=\mathrm{Z}\mathrm{i}\mathrm{n}\mathrm{b}$, 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.