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 ≻ 𝑏 = 𝑑(𝑎)𝑏, 𝑎 ≺ 𝑏 = 𝑎𝑑(𝑏), 𝑎, 𝑏 ∈ 𝐴. Given a variety Var of algebras, its derived variety is generated by all derived algebras 𝐴⁽ᵈ⁾ for all 𝐴 in 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 Var = 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.