Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 017, 15 pages      arXiv:2305.07371      https://doi.org/10.3842/SIGMA.2024.017

On Pre-Novikov Algebras and Derived Zinbiel Variety

Pavel Kolesnikov ^a, Farukh Mashurov ^b and Bauyrzhan Sartayev ^cd
a) Sobolev Institute of Mathematics, Novosibirsk, Russia
^b) Shenzhen International Center for Mathematics (SICM), Southern University of Science and Technology, Shenzhen, Guangdong, P.R. China
^c) Narxoz University, Almaty, Kazakhstan
^d) United Arab Emirates University, Al Ain, United Arab Emirates

Received August 31, 2023, in final form February 05, 2024; Published online February 28, 2024

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.

Key words: Novikov algebra; derivation; dendriform algebra; Zinbiel algebra.

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