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On Pre-Novikov Algebras and Derived Zinbiel Variety
Symmetry, integrability and geometry: methods and applications, Tome 20 (2024) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: Novikov algebra, derivation, dendriform algebra
Mots-clés : Zinbiel algebra. 
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}
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Pavel Kolesnikov; Farukh Mashurov; Bauyrzhan Sartayev. On Pre-Novikov Algebras and Derived Zinbiel Variety. Symmetry, integrability and geometry: methods and applications, Tome 20 (2024). http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a16/

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