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On Pre-Novikov Algebras and Derived Zinbiel Variety
Pavel Kolesnikova, Farukh Mashurovb, Bauyrzhan Sartayevcd 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 United Arab Emirates University, Al Ain, United Arab Emirates
d Narxoz University, Almaty, Kazakhstan
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.
Keywords:
Novikov algebra, derivation, dendriform algebra, Zinbiel algebra.
Received: August 31, 2023; in final form February 5, 2024; Published online February 28, 2024
Citation:
Pavel Kolesnikov, Farukh Mashurov, Bauyrzhan Sartayev, “On Pre-Novikov Algebras and Derived Zinbiel Variety”, SIGMA, 20 (2024), 017, 15 pp.
Linking options:
https://www.mathnet.ru/eng/sigma2019 https://www.mathnet.ru/eng/sigma/v20/p17
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