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B. K. Sartayev

Publications and source records attributed to B. K. Sartayev.

16 recordsLinked to original sources
arXiv2026

Identities in differential perm algebras

Let $(P,\cdot,d)$ be a differential perm algebra over a field of characteristic zero, i.e. an associative algebra satisfying $(ab)c=(ba)c$ and equipped with a derivation $d$. We study polynomial identities in the algebras obtained by the derived operations \[ a\prec b=ab',\quad a\succ b=a'b,\quad a\blacklozenge b=ab'+ba',\quad a\bullet b=a'b+ab',\quad a\Diamond b=ab'-ba',\quad a\circ b=a'b-ab', \] where $a'=d(a)$. We first prove that any nontrivial differential polynomial identity that does not belong to the right annihilator of the free differential perm algebra implies a differential identity of the form $a_1'a_2'\cdots a_m'=0$ for some positive integer $m$. We then obtain explicit generating sets and determine the dimensions of the multilinear homogeneous components of the subalgebras of the free differential perm algebra generated by $X$ with respect to the products $\blacklozenge$ and $\bullet$. Finally, we construct perm-Witt type Lie and Leibniz algebras arising naturally from differential perm algebras.

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arXiv2026

A vanishing criterion for Lie elements in a free Novikov algebra

Every Novikov algebra is Lie-admissible: the commutator turns it into a Lie algebra. We give a finite criterion for a multilinear element of the free Novikov algebra to belong to the Lie subalgebra generated by the free generators. By the differential realization of free Novikov algebras, the multilinear component of degree $n$ is identified with the space of homogeneous polynomials of degree $n-1$ in $n$ variables. We prove that a multilinear element is a Lie element if and only if its symbol vanishes at every integer point $(a_1,\ldots,a_n)$ with $a_i\le 1$ and $a_1+\cdots+a_n\ge 2$. Equivalently, the symbol is annihilated by two explicit linear differential operators of orders two and three. The proof combines the Witt algebra, a specialization argument for a symmetric block of variables, homogeneous interpolation and Molev's description of the multilinear component of the Lie algebra generated by the free generators as a module over the symmetric group. As applications we show that a nonzero multilinear Lie element is never a total derivative, recover the dimension of the multilinear Lie component, and illustrate the criterion by examples in degrees $4$, $5$ and $6$.

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arXiv2026

Initial pre-algebras as a generalization of dendriform algebras

We continue the study of \emph{initial dialgebras} defined in~\cite{DMS2026}. For a binary operad $\Var$ we define the class of initial pre-$\Var$-algebras and the corresponding operad $\pre\Var^{\I}$ in such a way that \[ (\di\Var^{\I})^{!}=(\pre(\Var^{!}))^{\I} \] in the case when $\Var $ is quadratic. We propose an intuitive algorithm for finding the defining relations of the operad $\pre\Var^{\I}$ in the case when $\Var$ is a binary quadratic operad. We also study free initial pre-algebras in the associative and commutative settings. For the nonsymmetric operad $\pre\As^{\I}$, we construct a Grobner--Shirshov basis in the free magma operad, describe a linear basis in terms of admissible decorated planar binary trees, and establish a bijection between these trees and certain combinatorial objects.

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arXiv2026

Nonsymmetric versions of binary quadratic operads

In this paper, we study the white Manin product of the associative operad $\As$ with a binary quadratic operad $\Var$. We introduce the notion of a nonsymmetric version of $\Var$ and provide a criterion for determining when the operad $\As\circ\Var$ has this property. We illustrate the construction with several examples and counterexamples. Finally, for some operads admitting nonsymmetric versions, we describe their combinatorial properties.

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arXiv2025

White Manin product and Hadamard product

In this paper, we consider three types of operads: alternative, assosymmetric, and bicommutative. We prove that the Hadamard product of these operads with the Novikov operad coincides with their white Manin product. As an application, we identify a variety of algebras in which all algebras are special.

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arXiv2024

On the Dong Property for a binary quadratic operad

The classical Dong Lemma for distributions over a Lie algebra lies in the foundation of vertex algebras theory. In this paper, we find necessary and sufficient condition for a variety of nonassociative algebras with binary operations to satisfy the analogue of the Dong Lemma. In particular, it turns out that Novikov and Novikov--Poisson algebras satisfy the Dong Lemma. The criterion is stated in the language of operads, so we determine for which binary quadratic operads the Dong Lemma holds true. As an application, we show the black Manin product of Dong operads is also a Dong operad.

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arXiv2024

Binary perm algebras and alternative algebras

In this paper, we describe the defining identities of a variety of binary perm algebras, which is a subvariety of the variety of alternative algebras. In addition, we construct a basis of the free binary perm algebra and find a complete list of identities which satisfy binary perm algebra under commutator.

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arXiv2024

On the free metabelian Novikov and metabelian Lie-admissible algebras

In this paper, we consider Lie-admissible algebras, which are free Novikov and free Lie-admissible algebras with an additional metabelian identity. We construct a linear basis for both free metabelian Novikov and free metabelian Lie-admissible algebras. Additionally, we describe a space of symmetric polynomials for both the free metabelian Novikov algebra and the free metabelian Lie-admissible algebra.

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arXiv2023

On the special identities of Gelfand--Dorfman algebras

In this paper, we prove that the class of all special Gelfand--Dorfman algebras (GD-algebras) is closed with respect to homomorphisms and thus forms a variety. We also prove that every 2-dimensional GD-algebra is special. For the latter, we give a technical method to find all special identities of GD-algebras and compute the degree 6 component of the Gröbner basis for the shuffle operad constructed on the symmetric operad governing the class of GD-algebras.

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arXiv2023

Some generalizations of the variety of transposed Poisson algebras

It is shown that the variety of transposed Poisson algebras coincides with the variety of Gelfand-Dorfman algebras in which the Novikov multiplication is commutative. The Gröbner-Shirshov basis for the transposed Poisson operad is calculated up to degree 4. Furthermore, we demonstrate that every transposed Poisson algebra is F-manifold. We verify that the special identities of GD-algebras hold in transposed Poisson algebras. Finally, we propose a conjecture stating that every transposed Poisson algebra is special, i.e., can be embedded into a differential Poisson algebra.

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arXiv2023

Basis of the free noncommutative Novikov algebra

As it is known, the defining identities of a free Novikov algebra can be obtained from a commutative algebra with a derivation. In this paper, we consider a class of algebras obtained from the class of associative algebras with a derivation that generalizes Novikov algebras. Such objects are called noncommutative Novikov algebras. We construct a monomial basis for a free noncommutative Novikov algebra.

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arXiv2022

Metabelian Lie and perm algebras

It is well known that any Lie algebra can be embedded into an associative algebra. We prove that any metabelian Lie algebra can be embedded into an algebra in the subvariety of perm algebras, i.e., associative algebras with the identity $abc-acb =0$. In addition, a technical method to construct the universal enveloping perm algebra for a metabelian Lie algebra is given.

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arXiv2022

Free special Gelfand-Dorfman algebra

A Gelfand-Dorfman algebra is called special if it can be embedded into a differential Poisson algebra. We find a new basis of the free Novikov algebra. With its help, we construct the monomial basis of the free special Gelfand-Dorfman algebra.

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arXiv2022

Novikov dialgebras and perm algebras

In this paper, we consider Perm algebra with the derivation $d$. The algebra itself is equipped with the new operation $a\succ b = d(a) b$. We construct a linear basis of the free Novikov dialgebra in terms of new operations. Also, we prove that the class of algebras under the new operation form a variety. Finally, we find the defining identities of the variety.

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arXiv2021

On the embedding of left-symmetric algebras into differential perm-algebras

Given an associative algebra satisfying the left commutativity identity $abc=bac$ (Perm-algebra) with a derivation $d$, the new operation $a\circ b = a d(b)$ is left-symmetric (pre-Lie). We derive necessary and sufficient conditions for a left-symmetric algebra to be embeddable into a differential Perm-algebra.

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