arXiv subjects

Valeriy G. Bardakov

Publications and source records attributed to Valeriy G. Bardakov.

At least 19 recordsLinked to original sources

arXiv2026

Multi-virtual braid groups

L. Kauffman (2024) introduced multi-virtual and symmetric multi-virtual braid groups, which are generalizations of the virtual braid group. We introduce multi-virtual pure and multi-virtual semi-pure braid groups, which are normal subgroups of index $n!$. We give a set of generators and defining relations for these groups, show that multi-virtual (symmetric multi-virtual) braid group is a semi-direct products of multi-virtual pure (symmetric multi-virtual pure) braid group and symmetric group. Also, we introduce multi-welded and multi-unrestricted braid groups and examines structure of three-strand 2-virtual braid group and some its subgroups and quotients. The paper concludes by outlining open problems and suggesting avenues for future research in this area.

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arXiv2025

Cayley graphs and their growth functions for multivalued groups

We define the Cayley graph and its growth function for multivalued groups. We prove that if we change a finite set of generators of multivalued group, or change the starting point, we get an equivalent growth function. We prove that if we take a virtually nilpotent group and construct a coset group with respect a finite group of authomorphisms, then this multivalued group has a polynomial growth. Also, we find a connection between this growth function and growth function of multivalued dynamics. It particular, it is obtained upper and lower bounds on growth functions of multivalued dynamics. We give a particular answer to a question of Buchstaber on polynomial growth of dynamics and a question of Buchstaber and Vesnin on growth functions of cyclically presented multivalued groups.

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arXiv2025

Self-distributive algebras and bialgebras

This article is devoted to the study of self-distributive algebraic structures: algebras, bialgebras; additional structures on them, relations of these structures with Hopf algebras, Lie algebras, Leibnitz algebras etc. The basic example of such structures are rack- and quandle bialgebras. But we go further - to the general coassociative comultiplication. The principal motivation for this work is the development of the linear algebra related with a notion of a quandle in analogy with the ubiquitous role of group algebras in the category of groups with perspective applications to the theory of knot invariants. We give description of self-distributive algebras and show that some quandle algebras and some Novikov algebras are self-distributive. Also, we give a full classification of counital self-distributive bialgebras in dimension 2 over C.

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arXiv2025

2-dimensional self-distributive non-counital bialgebras and knot invariants

In the preprint of V. Bardakov, T. Kozlovskaya, D. Talalaev (Self-distributive bialgebras, arXiv:2501.19152) it was formulated a problem of classification of self-distributive bialgebras and was given classification of two-dimensional counital self-distributive bialgebras. In this paper, we consider non-counital case. We find all 2-dimensional algebras of this type. In constructed algebras we study the question of finding quandles for constructing knot invariants. This activity is part of the overall program for the linearization of the concepts of rack and quandle and the development of the representations theory of these structures.

math.QA↗

arXiv2024

Algebra, group, and Hopf Rota-Baxter operators

We know definition of Rota--Baxter operators on different algebraic systems. For examples, on groups, on algebras, on Hopf algebras. On some algebraic systems it is possible to define different types of Rota--Baxter operators. For example, on group algebra it is possible to define Rota--Baxter operator as on associative algebra, group Rota--Baxter operator and Rota--Baxter operator as on a Hopf algebra. We are investigating the following question: What are connections between these operators? We are studying these questions for the Sweedler algebra $H_4$, that is 4-dimension non--cocommutative Hopf algebra.

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arXiv2024

Lie Rota--Baxter operators on the Sweedler algebra $H_4$

If $A$ is an associative algebra, then we can define the adjoint Lie algebra $A^{(-)}$ and Jordan algebra $A^{(+)}$. It is easy to see that any associative Rota--Baxter operator on $A$ induces a Lie and Jordan Rota--Baxter operator on $A^{(-)}$ and $A^{(+)}$ respectively. Are there Lie (Jordan) Rota--Baxter operators, which are not associative Rota--Baxter operators? In the present article we are studying these questions for the Sweedler algebra $H_4$, that is a 4-dimension non-commutative Hopf algebra. More precisely, we describe the Rota--Baxter operators on Lie algebra on the adjoint Lie algebra $H_4^{(-)}$.

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arXiv2024

Extensions of braid group representations to the monoid of singular braids

Given a representation $φ\colon B_n \to G_n$ of the braid group $B_n$, $n \geq 2$ into a group $G_n$, we are considering the problem of whether it is possible to extend this representation to a representation $Φ\colon SM_n \to A_n$, where $SM_n$ is the singular braid monoid and $A_n$ is an associative algebra, in which the group of units contains $G_n$. We also investigate the possibility of extending the representation $Φ\colon SM_n \to A_n$ to a representation $\widetildeΦ \colon SB_n \to A_n$ of the singular braid group $SB_n$. On the other hand, given two linear representations $φ_1, φ_2 \colon H \to GL_m(\Bbbk)$ of a group $H$ into a general linear group over a field $\Bbbk$, we define the defect of one of these representations with respect to the other. Furthermore, we construct a linear representation of $SB_n$ which is an extension of the Lawrence-Krammer-Bigelow representation (LKBR) and compute the defect of this extension with respect to the exterior product of two extensions of the Burau representation. Finally, we discuss how to derive an invariant of classical links from the Lawrence-Krammer-Bigelow representation.

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arXiv2023

Brunnian planar braids and simplicial groups

Twin groups are planar analogues of Artin braid groups and play a crucial role in the Alexander-Markov correspondence for the isotopy classes of immersed circles on the 2-sphere without triple and higher intersections. These groups admit diagrammatic representations, leading to maps obtained by the addition and deletion of strands. This paper explores Brunnian twin groups, which are subgroups of twin groups composed of twins that become trivial when any of their strands are deleted. We establish that Brunnian twin groups consisting of more than two strands are free groups. Furthermore, we provide a necessary and sufficient condition for a Brunnian doodle on the 2-sphere to be the closure of a Brunnian twin. Additionally, we delve into two generalizations of Brunnian twins, namely, $k$-decomposable twins and Cohen twins, and prove some structural results about these groups. We also investigate a simplicial structure on pure twin groups that admits a simplicial homomorphism from Milnor's construction of the simplicial 2-sphere. This gives a possibility to provide a combinatorial description of homotopy groups of the 2-sphere in terms of pure twins.

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arXiv2023

n-valued quandles and associated bialgebras

The principal aim of this article is to introduce and study n-valued quandles and n-corack bialgebras. We elaborate the basic methods of this theory, reproduce the coset construction known in the theory of n-valued groups. We also consider a construction of n-valued quandles using n-multi-quandles. In contrast to the case of n-valued groups this construction turns out to be quite rich in algebraic and topological applications. An important part of the work is the study of the properties of n-corack bialgebras those role is analogous to the group bialgebra.

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arXiv2023

Relative Rota--Baxter operators on groups and Hopf algebras

M. Goncharov introduced and studied a Rota--Baxter operator on a cocommutative Hopf algebra. In the present paper we define relative Rota--Baxter operators on an arbitrary Hopf algebra. A particular case of this definition is Goncharov's operator. On a Hopf algebra with a relative Rota--Baxter operator we define new associative operation and construct a new Hopf algebra and Hopf brace. Further, we construct Rota--Baxter operators of integer weights on some groups. The question on a possibility to define operator of zero weight on groups was formulated by X. Gao, L. Guo, Y. Liu, and Z.-C. Zhu. In the last section we construct a family of two generated Hopf algebras. This family includes some known Hopf algebras, in particular, 4-dimensional Sweedler algebra $H_4$.

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arXiv2023

Twisted Virtual Braid Group

In this paper we study some subgroups and their decompositions in semi-direct product of the twisted virtual braid group $TVB_n$. In particular, the twisted virtual pure braid group $TVP_n$ is the kernel of an epimorphism of $TVB_n$ onto the symmetric group $S_n$. We find the set of generators and defining relations for $TVP_n$ and show that $TVB_n = TVP_n \rtimes S_n$. Further we prove that $TVP_n$ is a semi-direct product of some subgroup and abelian group $\mathbb{Z}_2^n$. As corollary we get that the virtual pure braid group $VP_n$ is a subgroup of $TVP_n$. Also, we construct some other epimorphism of $TVB_n$ onto $S_n$. Its kernel, $TVH_n$ is an analogous of $TVP_n$. We find its set of generators and defining relations and construct its decomposition in a semi-direct product.

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arXiv2023

Multivalued groups and Newton polyhedron

On the set of complex number $\mathbb{C}$ it is possible to define $n$-valued group for any positive integer $n$. The $n$-multiplication defines a symmetric polynomial $p_n = p_n(x, y, z)$ with integer coefficients. By the theorem on symmetric polynomials, one can present $p_n$ as polynomial in elementary symmetric polynomials $e_1$, $e_2$, $e_3$. V.~M.~Buchstaber formulated a question on description coefficients of this polynomial. Also, he formulated the next question: How to describe the Newton polyhedron of $p_n$? In the present paper we find all coefficients of $p_n$ under monomials of the form $e_1^i e_2^j$ and prove that the Newton polyhedron of $p_n$ is an right triangle.

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arXiv2023

Symmetric skew braces and brace systems

For a skew left brace $(G, \cdot, \circ)$, the map $λ: (G, \circ) \to \Aut \,(G, \cdot),~~a \mapsto λ_a,$ where $λ_a(b) = a^{-1} \cdot (a \circ b)$ for all $a, b \in G$, is a group homomorphism. Then $λ$ can also be viewed as a map from $(G, \cdot)$ to $\Aut \, (G, \cdot)$, which, in general, may not be a homomorphism. A skew left brace will be called $λ$-anti-homomorphic ($λ$-homomorphic) if $λ: (G, \cdot) \to \Aut \, (G, \cdot)$ is an anti-homomorphism (a homomorphism). We mainly study such skew left braces. We device a method for constructing a class of binary operations on a given set so that the set with any two such operations constitute a $λ$-homomorphic symmetric skew brace. Most of the constructions of symmetric skew braces dealt with in the literature fall in the framework of our construction. We then carry out various such constructions on specific infinite sets.

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arXiv2022

Singular braids, singular links and subgroups of camomile type

In this paper we find a finite set of generators and defining relations for the singular pure braid group $SP_n$, $n \geq 3$, that is a subgroup of the singular braid group $SG_n$. Using this presentation, we prove that the center of $SG_n$ (which is equal to the center of $SP_n$ for $n \geq 3$) is a direct factor in $SP_n$ but it is not a direct factor in $SP_n$. We introduce subgroups of camomile type and prove that the singular pure braid group $SP_n$, $n \geq 5$, is a subgroup of camomile type in $SG_n$. Also we construct the fundamental singquandle using a representation of the singular braid monoid by endomorphisms of free guandle. For any singular link we define some family of groups which are invariants of this link.

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arXiv2022

Extensions of Yang-Baxter sets

The paper extends the notion of braided set and its close relative - the Yang-Baxter set - to the category of vector spaces and explore structure aspects of such a notion as morphisms and extensions. In this way we describe a family of solutions for the Yang-Baxter equation on the product of B and C if given B and C correspond to two linear (set-theoretic) solutions of the Yang-Baxter equation. One of the key observation is the relation of this question with the virtual pure braid group.

math.QA↗

arXiv2022

Multiplication of quandle structures

We generalise the construction of $Q$-family of quandles and $G$-family of quandles which were introduced in the paper of A. Ishii, M. Iwakiri, Y. Jang, K. Oshiro, and find connection with other constructions of quandles. We define a composition of quandl's structures, which are defined on the same set and find conditions under which this composition gives a quandle. Further we prove that under this multiplication we get a group and show that this group is abelian.

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arXiv2022

Rota-Baxter groups, skew left braces, and the Yang-Baxter equation

Braces were introduced by W. Rump in 2006 as an algebraic system related to the quantum Yang-Baxter equation. In 2017, L. Guarnieri and L. Vendramin defined for the same purposes a more general notion of a skew left brace. Recently, L. Guo, H. Lang, Y. Sheng [arXiv:2009.03492] gave a definition of what is a Rota-Baxter operator on a group. We connect these two notions as follows. It is shown that every Rota-Baxter group gives rise to a skew left brace. Moreover, every skew left brace can be injectively embedded into a Rota-Baxter group. When the additive group of a skew left brace is complete, then this brace is induced by a Rota-Baxter group. We interpret some notions of the theory of skew left braces in terms of Rota-Baxter operators.

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arXiv2021

Virtually Symmetric representations and marked Gauss diagrams

In this paper, we define the notion of a virtually symmetric representation of representations of virtual braid groups and prove that many known representations are equivalent to virtually symmetric. Using one such representation, we define the notion of virtual link groups which is an extension of virtual link groups defined by Kauffman. Moreover, we introduce the concept of marked Gauss diagrams as a generalisation of Gauss diagrams and their interpretation in terms of knot-like diagrams. We extend the definition of virtual link groups to marked Gauss diagrams and define their peripheral structure. We define $C_m$-groups and prove that every group presented by a $1$-irreducible $C_1$-presentation of deficiency $1$ or $2$ can be realized as the group of a marked Gauss diagram.

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