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P. S. Kolesnikov

Publications and source records attributed to P. S. Kolesnikov.

At least 19 recordsLinked to original sources
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

Chiral algebras with abelian conformal part

We study a categorical approach to the concept of varieties of chiral algebras. We prove that the class of chiral algebras in the variety defined by a binary quadratic operad Var, whose conformal structure is abelian, coincides with the class of differential algebras in the variety defined by the Manin black product of the operads Var and Com, where Com is the operad of associative commutative algebras.

math.QA↗
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

Differential envelopes of Novikov conformal algebras

A Novikov conformal algebra is a conformal algebra such that its coefficient algebra is right-symmetric and left commutative (i.e., it is an ``ordinary'' Novikov algebra). We prove that every Novikov conformal algebra with a uniformly bounded locality function on a set of generators can be embedded into a commutative conformal algebra with a derivation. In particular, every finitely generated Novikov conformal algebra has a commutative conformal differential envelope. For infinitely generated algebras this statement is not true in general.

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arXiv2024

On the locality of formal distributions over pre-Lie and Novikov algebras

The Dong Lemma in the theory of vertex algebras states that the locality property of formal distributions over a Lie algebra is preserved under the action of a vertex operator. A~similar statement is known for associative algebras. We study local formal distributions over pre-Lie (right-symmetric), pre-associative (dendriform), and Novikov algebras to show that the analogue of the Dong Lemma holds for Novikov algebras but does not hold for pre-Lie and pre-associative ones.

math.QA↗
arXiv2023

Groebner--Shirshov bases method for vertex algebras

In this note we show how to apply the Gröbner--Shirshov bases (GSB) method for modules over an associative algebra to the study of vertex algebras defined by generators and relations. We compute GSBs for a series of vertex algebras and study the problem of embedding of a left-symmetric algebra into a vertex one preserving the normally ordered product.

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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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arXiv2021

Quadratic Lie conformal superalgebras related to Novikov superalgebras

We study quadratic Lie conformal superalgebras associated with No\-vikov superalgebras. For every Novikov superalgebra $(V,\circ)$, we construct an enveloping differential Poisson superalgebra $U(V)$ with a derivation $d$ such that $u\circ v = ud(v)$ and $\{u,v\} = u\circ v - (-1)^{|u||v|} v\circ u$ for $u,v\in V$. The latter means that the commutator Gelfand--Dorfman superalgebra of $V$ is special. Next, we prove that every quadratic Lie conformal superalgebra constructed on a finite-dimensional special Gel'fand--Dorfman superalgebra has a finite faithful conformal representation. This statement is a step toward a solution of the following open problem: whether a finite Lie conformal (super)algebra has a finite faithful conformal representation.

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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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arXiv2021

On dimension theory of supermodules, super-rings and superschemes

We introduce the notion of Krull super-dimension of supermodules over certain super-commutative Noetherian super-rings. We investigate how this notion relates to the notion of odd regular sequence introduced by T.Schmitt and how it behaves with respect to the transition to the graded and bigraded supermodules and super-rings associated with the original ones. We also apply these results to the super-dimension theory of superschemes of finite type and their morphisms.

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arXiv2020

Defining relations and Gröbner--Shirshov bases of Poisson algebras as of conformal modules

We study the relation between Poisson algebras and representations of Lie conformal algebras. We establish a setting for the calculation of a Gröbner--Shirshov basis in a module over an associative conformal algebra and apply this technique to Poisson algebras considered as conformal modules over appropriate associative envelopes of current Lie conformal algebras. As a result, we obtain a setting for the calculation of a Gröbner--Shirshov basis in a Poisson algebra.

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arXiv2019

Gelfand--Dorfman algebras, derived identities, and the Manin product of operads

Gelfand--Dorfman bialgebras (GD-algebras) are nonassociative systems with two bilinear operations satisfying a series of identities that express Hamiltonian property of an operator in the formal calculus of variations. The paper is devoted to the study of GD-algebras related with differential Poisson algebras. As a byproduct, we obtain a general description of identities that hold for operations $a\succ b = d(a)b$ and $a\prec b = ad(b)$ on a (non-associative) differential algebra with a derivation~$d$.

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arXiv2019

Universal enveloping Poisson conformal algebras

Lie conformal algebras are useful tools for studying vertex operator algebras and their representations. In this paper, we establish close relations between Poisson conformal algebras and representations of Lie conformal algebras. We also calculate explicitly Poisson conformal brackets on the associated graded conformal algebras of universal associative conformal envelopes of Virasoro conformal algebra and Neveu--Schwartz conformal superalgebra.

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arXiv2018

Derived identities of differential algebras

Suppose $A$ is a not necessarily associative algebra with a derivation $d$. Then $A$ may be considered as a system with two binary operations $\succ $ and $\prec $ defined by $x\succ y = d(x)y$, $x\prec y = xd(y)$, $x,y\in A$. Suppose $A$ satisfies some multi-linear polynomial identities. We show how to find the identities that hold for operations $\prec $ and $\succ $. It turns out that if $A$ belongs to a variety governed by an operad Var then $\succ $ and $\prec $ satisfy the defining relations of the operad Var$\circ $Nov, where $\circ $ is the Manin white product of operads, Nov is the operad of Novikov algebras. Moreover, there are no other identities that hold for operations $\succ $, $\prec $ on an arbitrary differential Var-algebra.

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arXiv2018

On the Hochschild cohomologies of associative conformal algebras with a finite faithful representation

Associative conformal algebras of conformal endomorphisms are of essential importance for the study of finite representations of conformal Lie algebras (Lie vertex algebras). We describe all semisimple algebras of conformal endomorphisms which have the trivial second Hochschild cohomology group with coefficients in every conformal bimodule. As a consequence, we state a complete solution of the radical splitting problem in the class of associative conformal algebras with a finite faithful representation.

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arXiv2016

Operads of decorated trees and their duals

This is an extended version of a talk presented by the second author on the Third Mile High Conference on Nonassociative Mathematics (August 2013, Denver, CO). The purpose of this paper is twofold. First, we would like to review the technique developed in a series of papers for various classes of di-algebras and show how do the same ideas work for tri-algebras. Second, we present a general approach to the definition of pre- and post-algebras which turns out to be equivalent to the construction of dendriform splitting. However, our approach is more algebraic and thus provides simpler way to prove various properties of pre- and post-algebras in general.

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