Searcharxiv⌕ Search

arXiv subjects

Pavel Kolesnikov

Publications and source records attributed to Pavel Kolesnikov.

At least 19 recordsLinked to original sources
arXiv2026

Cohomology theory of Novikov algebras and applications

In this paper, first we give a new characterization of the cohomology of pre-Lie algebras using the Chevalley-Eilenberg cohomology associated to a morphism from the operad of Lie algebras to Hadamard product of the operad of pre-Lie algebras and its Koszul dual operad. Then we apply the same approach to study the cohomology of Novikov algebras, and give the cochain complex explicitly. The cochain complex of the underlying pre-Lie algebra is shown to be isomorphic to the quotient of the cochain complex of a Novikov algebra. Consequently, there is a long exact sequence connecting the cohomologies of a Novikov algebra and the underlying pre-Lie algebra. The cohomology of a Novikov algebra with coefficients in a representation is introduced using pseudo-tensor categories. As applications, we show that infinitesimal deformations and abelian extensions are classified by the second cohomology groups with different coefficients. Various examples are given to illustrate the difference between the cohomology of a Novikov algebra and that of the underlying pre-Lie algebra.

math.RA↗
arXiv2026

On the structure of finite Novikov conformal algebras

We classify finite simple Novikov conformal algebras over an algebraically closed field of characteristic zero. The classification consists of the current conformal algebra over the base field and a one-parameter list of Virasoro-like conformal algebras. A semisimple finite Novikov conformal algebra is proved to be a direct sum of simple ones. We also describe deformations and central extensions of finite simple Novikov conformal algebras.

math.RA↗
arXiv2024

Hochschild cohomology of the universal associative conformal envelope of the Virasoro Lie conformal algebra with coefficients in all finite modules

In this paper, we find the Hochschild cohomology groups of the universal associative conformal envelope $U(3)$ of the Virasoro Lie conformal algebra with respect to associative locality $N=3$ on the generator with coefficients in all finite modules. In order to obtain this result, we construct the Anick resolution via the algebraic discrete Morse theory and Gröbner--Shirshov basis.

math.RA↗
arXiv2024

Formal integration of complete Rota-Baxter Lie algebras

In this paper, first we revisit the formal integration of Lie algebras, which give rise to braces in some special cases. Then we establish the formal integration theory for complete Rota-Baxter Lie algebras, that is, we show that there is a Rota-Baxter group with the underlying group structure given by the Baker-Campbell-Hausdorff formula, associated to any complete Rota-Baxter Lie algebra. In particular, we use the post-Lie Magnus expansion to give the explicit formula of the Rota-Baxter operator. Finally we show that one can obtain a graded Rota-Baxter Lie ring from a filtered Rota-Baxter group.

math-ph↗
arXiv2024

On Pre-Novikov Algebras and Derived Zinbiel Variety

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.

math.RA↗
arXiv2021

Homogeneous conformal averaging operators on semisimple Lie algebras

In this note we show a close relation between the following objects: Classical Yang -- Baxter equation (CYBE), conformal algebras (also known as vertex Lie algebras), and averaging operators on Lie algebras. It turns out that the singular part of a solution of CYBE (in the operator form) on a Lie algebra $\mathfrak g$ determines an averaging operator on the corresponding current conformal algebra $\mathrm{Cur} \mathfrak g$. For a finite-dimensional semisimple Lie algebra $\mathfrak g$, we describe all homogeneous averaging operators on $\mathrm{Cur}\mathfrak g$. It turns out that all these operators actually define solutions of CYBE with a pole at the origin.

math.QA↗
arXiv2018

The Hochschild cohomology of the group $G^2_3$

We apply discrete algebraic Morse theory to calculate the Anick resolution of the group algebra of the group $G_3^2$. As a corollary, we evaluate Hochschild cohomologies of $G_3^2$ with coefficients in all 1-dimensional bimodules. Almost all these groups are trivial, the only exceptions are 1-dimensional $H^2$ for two particular 1-dimensional bimodules.

math.RA↗
arXiv2017

Groebner---Shirshov bases for replicated algebras

We establish a universal approach to solution of the word problem in the varieties of di- and tri-algebras. This approach, for example, allows to apply Groebner---Shirshov bases method for Lie algebras to solve the ideal membership problem in free Leibniz algebras (Lie di-algebras). As another application, we prove an analogue of the Poincare---Birkhoff---Witt Theorem for universal enveloping associative tri-algebra of a Lie tri-algebra (CTD^!-algebra).

math.RA↗
arXiv2016

Groebner-Shirshov basis of the universal enveloping Rota-Baxter algebra of a Lie algebra

Consider the class RBLie of Lie algebras equipped with a Rota---Baxter operator. Then the forgetful functor RBLie --> Lie has a left adjoint one denoted by $U_{RB}(\cdot)$. We prove an "operator" analogue of the Poincare---Birkhoff---Witt theorem for $U_{RB}(L)$, where $L$ is an arbitrary Lie algebra, by means of Gröbner---Shirshov bases theory for Lie algebras with an additional operator.

math.QA↗
arXiv2011

Graded associative conformal algebras of finite type

In this paper, we consider graded associative conformal algebras. The class of these objects includes pseudo-algebras over non-cocommutative Hopf algebras of regular functions on some linear algebraic groups. In particular, an associative conformal algebra which is graded by a finite group $Γ$ is a pseudo-algebra over the coordinate Hopf algebra of a linear algebraic group $G$ such that the identity component $G^0$ is the affine line and $G/G^0\simeq Γ$. A classification of simple and semisimple graded associative conformal algebras of finite type is obtained.

math.QA↗
arXiv2011

On representations of dialgebras and conformal algebras

In this note, we observe a relation between dialgebras (in particular, Leibniz algebras) and conformal algebras. The purpose is to show how the methods of conformal algebras help solving problems on dialgebras, and, conversely, how the ideas of dialgebras work for conformal algebras.

math.QA↗
arXiv2010

On finite representations of conformal algebras

We prove a finite torsion-free associative conformal algebra to have a finite faithful conformal representation. As a corollary, it is shown that one may join a conformal unit to such an algebra. Some examples are stated to demonstrate that a conformal unit can not be joined to any torsion-free associative conformal algebra. In particular, there exist associative conformal algebras of linear growth and even locally finite ones that have no finite faithful representation. We also consider the problem of existence of a finite faithful representation for a torsion-free finite Lie conformal algebra (the analogue of Ado's Theorem). It turns out that the conformal analogue of the Poincare-Birkhoff-Witt Theorem would imply the Ado Theorem for finite Lie conformal algebras. We prove that every torsion-free finite solvable Lie conformal algebra has a finite faithful representation.

math.QA↗
arXiv2009

Simple Finite Jordan Pseudoalgebras

We consider the structure of Jordan $H$-pseudoalgebras which are linearly finitely generated over a Hopf algebra $H$. There are two cases under consideration: $H=U(\mathfrak h)$ and $H=U(\mathfrak h)# \mathbb C[Γ]$, where $\mathfrak h$ is a finite-dimensional Lie algebra over $\mathbb C$, $Γ$ is an arbitrary group acting on $U(\mathfrak h)$ by automorphisms. We construct an analogue of the Tits-Kantor-Koecher construction for finite Jordan pseudoalgebras and describe all simple ones.

math.QA↗