# Valeriy BardakovSobolev Institute of Mathematics, Russian Academy of Scinces, Russia, Novosibirsk · Inverse problems of Mathematical Phisics

Valeriy Bardakov

Professor

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160

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Citations since 2017

## Publications

Publications (160)

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 $e_1$, $e_2$, $e_3$....

Сделан первый шаг в построении категории сплетеных множеств и ее ближайшего родственника - категории множеств Янга-Бакстера. Основной акцент делается на построении морфизмов и расширений множеств Янга-Бакстера. Важность такой проблемы обусловлена возможностью построения новых решений уравнения Янга-Бакстера и уравнения косы. Основным результатом яв...

Knots naturally appear in continuous dynamical systems as flow periodic trajectories. However, discrete dynamical systems are also closely connected with the theory of knots and links. For example, for Pixton diffeomorphisms, the equivalence class of the Hopf knot, which is the orbit space of the unstable saddle separatrix in the manifold $\mathbb{...

For a skew left brace ( G , ⋅ , ∘ ) {(G,\cdot\,,\circ)} , the map λ : ( G , ∘ ) → Aut ( G , ⋅ ) {\lambda:(G,\circ)\to\operatorname{Aut}(G,\cdot\,)} , a ↦ λ a , {a\mapsto\lambda_{a},} where λ a ( b ) = a - 1 ⋅ ( a ∘ b ) {\lambda_{a}(b)=a^{-1}\cdot(a\circ b)} for all a , b ∈ G {a,b\in G} , is a group homomorphism. Then λ can also be viewed as a m...

The theory of Rota–Baxter operators on rings and algebras has been developed since 1960. In 2020, the notion of Rota–Baxter operator on a group was defined. Further, it was proved that one may define a skew left brace on any group endowed with a Rota–Baxter operator. Thus, a group endowed with a Rota–Baxter operator gives rise to a set-theoretical...

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 f...

We introduce the notion of a (virtual) multi-switch which generalizes the notion of a (virtual) switch. Using (virtual) multi-switches we introduce a general approach of how to construct representations of (virtual) braid groups by automorphisms of algebraic systems. As a corollary we introduce new representations of virtual braid groups which gene...

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-th...

The $n$-simplex equation ($n$-SE) was introduced by A. B. Zamolodchikov as a generalization of the Yang--Baxter equation, which is the $2$-simplex equation in these terms. In the present paper we suggest some general approaches to constructing solutions of $n$-simplex equations, describe some types of solutions, introduce an operation which under s...

Cycle sets are known to give non-degenerate unitary solutions of the Yang-Baxter equation and linear cycle sets are enriched versions of these algebraic systems. The paper explores the recently developed cohomology and extension theory for linear cycle sets. We derive a four term exact sequence relating 1-cocycles, second cohomology and certain gro...

The paper develops further the theory of quandle rings which was introduced by the authors in a recent work. Orderability of quandles is defined and many interesting examples of such quandles are given. It is proved that quandle rings of left or right orderable quandles which are semi-latin have no zero-divisors. Idempotents in quandle rings of cer...

For a skew left brace $(G, \cdot, \circ)$, the map $\lambda : (G, \circ) \to \Aut \,(G, \cdot),~~a \mapsto \lambda_a,$ where $\lambda_a(b) = a^{-1} \cdot (a \circ b)$ for all $a, b \in G$, is a group homomorphism. Then $\lambda$ can also be viewed as a map from $(G, \cdot)$ to $\Aut \, (G, \cdot)$, which, in general, may not be a homomorphism. A sk...

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 compositi...

In the present paper, we study structural aspects of certain quotients of braid groups and virtual braid groups. In particular, we construct and study linear representations B n → GL n ( n - 1 ) / 2 ( Z [ t ± 1 ] ) , VB n → GL n ( n - 1 ) / 2 ( Z [ t ± 1 , t 1 ± 1 , t 2 ± 1 , … , t n - 1 ± 1 ] ) which are connected with the famous Lawre...

The homotopy braid group B ^ n {\widehat{B}_{n}} is the subject of the paper. First, the linearity of B ^ n {\widehat{B}_{n}} over the integers is proved. Then we prove that the group B ^ 3 {\widehat{B}_{3}} is torsion free.

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. 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...

In this paper, we show that every finite spatial graph is a connected sum of a planar graph, which is a forest, and a tangle. As a consequence, we get that any finite spatial graph is a connected sum of a planar graph and a braid. Using these decompositions it is not difficult to find a set of generators and defining relations for the fundamental g...

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,...

For a skew left brace (G,⋅,∘), the map λ:(G,∘)→Aut(G,⋅),a↦λa, where λa(b)=a−1⋅(a∘b) for all a,b∈G, is a group homomorphism. Then λ can also be viewed as a map from (G,⋅) to Aut(G,⋅), which, in general, may not be a homomorphism. We study skew left braces (G,⋅,∘) for which λ:(G,⋅)→Aut(G,⋅) is a homomorphism. Such skew left braces will be called λ-ho...

Biquandles are algebraic objects with two binary operations whose axioms encode the generalized Reidemeister moves for virtual knots and links. These objects also provide set theoretic solutions of the well-known Yang-Baxter equation. The first half of this paper proposes some natural constructions of biquandles from groups and from their simpler c...

In the present paper we study structural aspects of certain quotients of braid groups and virtual braid groups. In particular, we construct and study linear representations $B_n\to {\rm GL}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm1}]\right)$, $VB_n\to {\rm GL}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm1}, t_1^{\pm1},t_2^{\pm1},\ldots, t_{n-1}^{\pm1}]\right)$ which...

The paper establishes new relationship between cohomology, extensions and automor-phisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an abelian extension of quandles. A non-abelian counterpart of this sequence involving dynamical cohomology classes is...

We study the following question: under what conditions extension of one residually nilpotent group by another residually nilpotent group is residually nilpotent? We prove some sufficient conditions under which this extension is residually nilpotent. Also, we study this question for semi-direct products and, in particular, for extensions of free gro...

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 conne...

Theory of Rota-Baxter operators on rings and algebras has been developed since 1960. Recently, L. Guo, H. Lang, Y. Sheng [arXiv:2009.03492] have defined the notion of Rota-Baxter operator on a group. We provide some general constructions of Rota-Baxter operators on a group. Given a map on a group, we study its extensions to a Rota-Baxter operator....

Cycle sets are known to give non-degenerate unitary solutions of the Yang--Baxter equation and linear cycle sets are enriched versions of these algebraic systems. The paper explores the recently developed cohomology and extension theory for linear cycle sets. We derive a four term exact sequence relating 1-cocycles, second cohomology and certain gr...

Given a virtual biquandle multi-switch (S,V) on an algebraic system X (from some category) and a virtual link L, we introduce a general approach to construct an algebraic system XS,V(L) (from the same category) which is an invariant of L. As a corollary we introduce a new quandle invariant for virtual links which generalizes the previously known qu...

In the paper, we construct a representation $\theta:FVB_n\to{\rm Aut}(F_{2n})$ of the flat virtual braid group $FVB_n$ on $n$ strands by automorphisms of the free group $F_{2n}$ with $2n$ generators which does not preserve the forbidden relations in the flat virtual braid group. This representation gives a positive answer to the problem formulated...

We find the lower central series for residually nilpotent Baumslag-Solitar groups, and find the intersection of all terms of the lower central series. Also, we find non-abelian Bauslag-Solitar groups for which the lower central series has length 2. For some Baumslag-Solitar groups a connection is found between the intersection of all subgroups of f...

We look at Baumslag–Solitar groups. Lower central series are given for residually nilpotent Baumslag–Solitar groups. For some Baumslag–Solitar groups that are not residually nilpotent, we find the intersection of all terms of the lower central series. Also we point out non-Abelian Baumslag–Solitar groups having lower central series of length 2. For...

Homotopy braid group is the subject of the paper. First, linearity of homotopy braid group over the integers is proved. Then we prove that the group homotopy braid group on three strands is torsion free.

In this paper, we study the singular pure braid group [Formula: see text] for [Formula: see text]. We find generators, defining relations and the algebraical structure of these groups. In particular, we prove that [Formula: see text] is a semi-direct product [Formula: see text], where [Formula: see text] is an HNN-extension with base group [Formula...

In this paper we define virtually symmetric representations of virtual braid group $VB_n$ and prove that many previously known representations are equivalent to virtually symmetric representations. Using a virtually symmetric representation we define virtual link group. We define and study group system of virtual knots by defining marked Gauss diag...

We study groups of some virtual knots with small number of crossings and prove that there is a virtual knot with long lower central series which, in particular, implies that there is a virtual knot with residually nilpotent group. This gives a possibility to construct invariants of virtual knots using quotients by terms of the lower central series...

The paper investigates exterior and symmetric (co)homologies of groups. We introduce symmetric homology of groups and compute exterior and symmetric (co)homologies of some finite groups. We also compare the classical, exterior and symmetric (co)homologies. Finally, we derive restriction and corestriction homomorphisms for exterior cohomology.

The paper investigates exterior and symmetric (co)homologies of groups. We introduce symmetric homology of groups and compute exterior and symmetric (co)homologies of some finite groups. We also compare the classical, exterior and symmetric (co)homologies. Finally, we derive restriction and corestriction homomorphisms for exterior cohomology.

This paper is dedicated to cabling on virtual braids. This construction gives a new generating set for the virtual pure braid group [Formula: see text]. We define simplicial group [Formula: see text] and its simplicial subgroup [Formula: see text] which is generated by [Formula: see text]. Consequently, we describe [Formula: see text] as HNN-extens...

In this paper, we introduce the new construction of quandles. For a group [Formula: see text] and a subset [Formula: see text] of [Formula: see text] we construct a quandle [Formula: see text] which is called the [Formula: see text]-quandle and study properties of this quandle. In particular, we prove that if [Formula: see text] is a quandle such t...

In this paper we show that every finite spatial graph is a connected sum of a planar graph, which is a forest, i.e. disjoint union of finite number of trees and a tangle. As a consequence we get that any finite spatial graph is a connected sum of a planar graph and a braid. Using these decompositions it is not difficult to find a set of generators...

In the present paper we study the singular pure braid group $SP_{n}$ for $n=2, 3$. We find generators, defining relations and the algebraical structure of these groups. In particular, we prove that $SP_{3}$ is a semi-direct product $SP_{3} = \widetilde{V}_3 \leftthreetimes \mathbb{Z}$, where $\widetilde{V}_3$ is an HNN-extension with base group $\m...

The paper establishes new relationship between cohomology, extensions and automorphisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an abelian extension of quandles. A non-abelian counterpart of this sequence involving dynamical cohomology classes is...

For a skew left brace $(G, \cdot, \circ)$, the map $\lambda : (G, \circ) \to \mathrm{Aut} \;(G, \cdot),~~a \mapsto \lambda_a$, where $\lambda_a(b) = a^{-1} \cdot (a \circ b)$ for all $a, b \in G$, is a group homomorphism. Then $\lambda$ can also be viewed as a map from $(G, \cdot)$ to $\mathrm{Aut}\; (G, \cdot)$, which, in general, may not be a hom...

Residual finiteness is known to be an important property of groups appearing in combinatorial group theory and low dimensional topology. In a recent work (Bardakov et al. in Proc Am Math Soc 147:3621–3633, 2019. https://doi.org/10.1090/proc/14488) residual finiteness of quandles was introduced, and it was proved that free quandles and knot quandles...

In this article we prove theorem on Lifting for the set of virtual pure braid groups. This theorem says that if we know presentation of virtual pure braid group $VP_4$, then we can find presentation of $VP_n$ for arbitrary $n > 4$. Using this theorem we find the set of generators and defining relations for simplicial group $T_*$ which was defined i...

The paper develops further the theory of quandle rings which was introduced by the authors in a recent work. Orderability of quandles is defined and many interesting examples of orderable quandles are given. It is proved that quandle rings of left or right orderable quandles which are semi-latin have no zero-divisors. Idempotents in quandle rings o...

In the paper we introduce a general approach how for a given virtual biquandle multi-switch $(S,V)$ on an algebraic system $X$ (from some category) and a given virtual link $L$ construct an algebraic system $X_{S,V}(L)$ (from the same category) which is an invariant of $L$. As a corollary we introduce a new quandle invariant for virtual links which...

We improve Algorithm 5.1 of [Math. Comp. 86 (2017) 2519–2534] for computing all nonisomorphic skew left braces, and enumerate left braces and skew left braces of orders up to 868 with some exceptions. Using the enumerated data, we state some conjectures for further research.

The twin group $T_n$ is a Coxeter group generated by $n-1$ involutions and the pure twin group $PT_n$ is the kernel of the natural surjection of $T_n$ onto the symmetric group on $n$ letters. In this paper, we investigate structural aspects of twin and pure twin groups. We prove that the twin group $T_n$ decomposes into a free product with amalgama...

Residual finiteness is known to be an important property of groups appearing in combinatorial group theory and low dimensional topology. In a recent work [2] residual finiteness of quandles was introduced, and it was proved that free quandles and knot quandles of tame knots are residually finite. In this paper, we extend these results and prove tha...

Biquandles are algebraic objects with two binary operations whose axioms encode the generalized Reidemeister moves for virtual knots and links. These objects also provide set-theoretic solutions of the well-known Yang-Baxter equation. The first half of this paper proposes some natural constructions of biquandles from groups and from their simpler c...

In the paper, we introduce the notion of a (virtual) multi-switch which generalizes the notion of a (virtual) switch. Using (virtual) multi-switches we introduce a general approach on how to construct representations of (virtual) braid groups by automorphisms of algebraic systems. As a corollary, we introduce new representations of virtual braid gr...

By improving Algorithm 5.1 of [Math. Comp. {\bf 86} (2017), 2519-2534], we enumerate left braces and skew left braces of orders upto 511 with some exceptions.

In this paper, we develop a theory of rack (respectively, quandle) rings analogous to the classical theory of group rings for groups. Let $X$ be a rack and $R$ an associative ring with unity. We introduce the rack ring $R[X]$ of $X$ with coefficients in $R$, and show that the ring $R[X]$ gives interesting information about the rack $X$. We define t...

This article is dedicate to cabling on virtual braids. This construction gives a new generating set for the virtual pure braid group $VP_n$. Consequently we describe $VP_4$ as HNN-extension. As an application to classical braids, we find a new presentation of the Artin pure braid group $P_4$ in terms of the cabled generators.

In this paper we study different questions concerning automorphisms of quandles. For a conjugation quandle $Q={\rm Conj}(G)$ of a group $G$ we determine several subgroups of ${\rm Aut}(Q)$ and find necessary and sufficient conditions when these subgroups coincide with the whole group ${\rm Aut}(Q)$. In particular, we prove that ${\rm Aut}({\rm Conj...

In this note, residual finiteness of quandles is defined and investigated. It is proved that free quandles and knot quandles of tame knots are residually finite and Hopfian. Residual finiteness of quandles arising from residually finite groups (conjugation, core and Alexander quandles) is established. Further, residual finiteness of automorphism gr...

We study groups of classical links, welded links, and virtual links. For classical braids, it is proved that the closures of a braid and its automorphic image are weakly equivalent. This implies the affirmative answer to the question of the coincidence of the groups constructed from a braid and from its automorphic image. We also study the problem...

In the present paper, we introduce the new construction of quandles. For a group $G$ and its subset $A$ we construct a quandle $Q(G,A)$ which is called the $(G,A)$-quandle and study properties of this quandle. In particular, we prove that if $Q$ is a quandle such that the natural map $Q\to G_Q$ from $Q$ to its enveloping group $G_Q$ is injective, t...

It is well known that for any classical knot group, the second term of the lower central series is equal to the third term of the lower central series. In this paper, we study groups of some virtual knots with small number of crossings and prove that there is a virtual knot with long lower central series which, in particular, implies that there is...

In this paper, we investigate exterior and symmetric (co)homology of groups. We give a new approach to symmetric cohomology and also introduce symmetric homology of groups. We compute symmetric homology and exterior (co)homology of some finite groups. Further, we compare the classical, exterior and symmetric (co)homology and introduce some new (co)...

For a pair of groups G, H, we study pairs of actions G on H and H on G such that these pairs are compatible. We prove that there are nilpotent group G and some group H such that for \(G \otimes H\) the derivative group [G, H] is equal to G. Also, we prove that if \(\mathbb {Z}_2\) act by inversion on an abelian group A, then the non-abelian tensor...

In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $n>3$, $\Aut(P_n)$ is generated by the subgroup $\Aut_c(P_n)$ of central automorphisms of $P_n$, the subgroup $\Aut(B_n)$ of restrictions of automorphisms of $B_n$ on $P_n$ and one extra automorphism $w_n$. We also investigate the lifting...

By exploring simplicial structure of pure virtual braid groups, we give new connections between the homotopy groups of the 3-sphere and the virtual braid groups that are related to the theory of Brunnian virtual braids. The group structure of VP_n with n > 4 is determined by VP_3, VP_4 and virtual cablings given by iterated degeneracy operations on...

In this note, residual finiteness of quandles is defined and investigated. It is proved that free quandles and knot quandles of tame knots are residually finite and Hopfian. Residual finiteness of quandles arising from residually finite groups (conjugation, core and Alexander quandles) is established. Further, residual finiteness of automorphism gr...

In the paper of Yu. A. Mikhalchishina for an arbitrary virtual link $L$ three groups $G_{1,r}(L)$, $r>0$, $G_{2}(L)$ and $G_{3}(L)$ were defined. In the present paper these groups for the virtual trefoil are investigated. The structure of these groups are found out and the fact that some of them are not isomorphic to each other is proved. Also we p...

Let $VB_n$, resp. $WB_n$ denote the virtual, resp. welded, braid group on $n$ strands. We study their commutator subgroups $VB_n' = [VB_n, VB_n]$ and, $WB_n' = [WB_n, WB_n]$ respectively. We obtain a set of generators and defining relations for these commutator subgroups. In particular, we prove that $VB_n'$ is finitely generated if and only if $n...

We study a representation of the virtual braid group VBn into the automorphism group of a free product of a free group and a free Abelian group, proposed by S. Kamada. It is proved that the given representation is equivalent to the representation constructed in [http://arxiv.org/abs/1603.01425]; i.e. the kernels of these representations coincide.

Let $G$ be a group and $\varphi \in \Aut(G)$. Then the set $G$ equipped with the binary operation $a*b=\varphi(ab^{-1})b$ gives a quandle structure on $G$, denoted by $\Alex(G, \varphi)$ and called the generalised Alexander quandle. When $G$ is additive abelian and $\varphi = -\id_G$, then $\Alex(G, \varphi)$ is the well-known Takasaki quandle. In...

In this paper we give an example of a linear group such that its tensor square is not linear. Also, we formulate some sufficient conditions for the linearity of non-abelian tensor products $G \otimes H$ and tensor squares $G \otimes G$. Using these results we prove that tensor squares of some groups with one relation and some knot groups are linear...

For a pair of groups $G, H$ we study pairs of actions $G$ on $H$ and $H$ on $G$ such that these pairs are compatible and non-abelian tensor products $G \otimes H$ are defined.

The authors have previously constructed two representations of the virtual braid group into the automorphism group of the free product of a free group and a free abelian group. Using them, we construct the two groups, each of which is a virtual link invariant. By the example of the virtual trefoil knot we show that the constructed groups are not is...

We study some subgroups of the automorphism group of a free group, their factorizations into a semidirect product, automorphism groups, and adjoint Lie algebras.

E. Rips constructed a series of groups such that their group rings have zero divisors. such groups can serve as counterexamples to Kaplansky’s problem on zero divisors. The problem is to find such a group without torsion. We study simplest groups of this series, classify such groups, describe the structure, and show that all such groups have 2-tors...

In this paper we study the kernel of the homomorphism $B_{g,n} \to B_n$ of the braid group $B_{g,n}$ in the handlebody $\mathcal{H}_g$ to the braid group $B_n$. We prove that this kernel is a semi-direct product of free groups. Also, we introduce an algebra $H_{g,n}(q)$, which is some analog of the Hecke algebra $H_n(q)$, constructed by the braid g...

Let $0 \to A \to L \to B \to 0$ be a short exact sequence of Lie algebras
over a field $F$, where $A$ is abelian. We show that the obstruction for a pair
of automorphisms in $Aut(A) \times Aut(B)$ to be induced by an automorphism in
$Aut(L)$ lies in the Chevalley-Eilenberg cohomology $H^2(B;A)$. As a
consequence, we obtain a four term exact sequenc...

In this paper, we initiate the study of palindromic automorphisms of groups that are free in some variety. More specifically, we define palindromic automorphisms of free nilpotent groups and show that the set of such automorphisms is a group. We find a generating set for the group of palindromic automorphisms of free nilpotent groups of step 2 and...

In the present paper the representation of the virtual braid group $VB_n$ into the automorphism group of free product of the free group and free abelian group is constructed. This representation generalizes the previously constructed ones. The fact that these already known representations are not faithful for $n \geq 4$ is verified. Using represent...

In this paper we define a monoid of pseudo braids and prove that this monoid
is isomorphic to a singular braid monoid. We also prove an analogue of Markov's
theorem for pseudo braids.

We consider the group of unrestricted virtual braids, describe its structure and explore its relations with fused links. Also, we define the groups of flat virtual braids and virtual Gauss braids and study some of their properties, in particular their linearity.

As is well known, every finite subgroup of the automorphism group of the polynomial algebra of rank two over a field of characteristic zero is conjugate to the subgroup of linear automorphisms. We show that this can fail for an arbitrary periodic subgroup. We construct an example of an Abelian p-subgroup of the automorphism group of the polynomial...

We consider the group of unrestricted virtual braids, describe its structure and explore its relations with fused links. Also, we define the group of flat virtual braids and the group of flat welded braids, and study some of their properties.

We present and discuss some open problems formulated by participants of the
International Workshop "Knots, Braids, and Auto\-mor\-phism Groups" held in
Novosibirsk, 2014. Problems are related to palindromic and commutator widths of
groups; properties of Brunnian braids and two-colored braids, corresponding to
an amalgamation of groups; extreme prop...

We prove that the nilpotent product of a set of groups $A_{1},\dots, A_{s}$
has finite palindromic width if and only if the palindromic widths of $A_{i},
i=1,\dots, s,$ are finite. We give a new proof that the commutator width of
$F_n \wr K$ is infinite, where $F_n$ is a free group of rank $n \geq 2$ and $K$
a finite group. This result, combining w...

It is well known that every finite subgroup of automorphism group of
polynomial algebra of rank 2 over the field of zero characteristic is
conjugated with a subgroup of linear automorphisms. We prove that it is not
true for an arbitrary torsion subgroup. We construct an example of abelian
$p$-group of automorphism of polynomial algebra of rank 2 ov...

We explore relationships between several different representations of braid groups as automorphisms of free groups as well as induced linear representations.

In this paper, we initiate the study of palindromic automorphisms of groups
that are free in some variety. More specifically, we define palindromic
automorphisms of free nilpotent groups and show that the set of such
automorphisms is a group. We find a generating set for the group of palindromic
automorphisms of free nilpotent groups of step 2 and...

We study some subgroups of the group of unitriangular automorphisms $U_n$ of
a free associative algebra over a field of characteristic zero. We find the
center of $U_n$ and describe the hypercenters of $U_2$ and $U_3$. In
particular, we prove that the upper central series for $U_2$ has infinite
length. As consequence, we prove that the groups $U_n$...

We investigate the palindromic width of finitely generated solvable groups.
We prove that every finitely generated $3$-step solvable group has finite
palindromic width. More generally, we show the finiteness of palindromic width
for finitely generated abelian-by-nilpotent-by-nilpotent groups. For arbitrary
solvable groups of step $\geq 3$, we prove...

In arXiv:1303.1129, the authors provided a bound for the palindromic width of
free abelian-by-nilpotent group $AN_n$ of rank $n$ and free nilpotent group
${\rm N}_{n,r}$ of rank $n$ and step $r$. In the present paper we study
palindromic widths of groups $\widetilde{AN}_n$ and $\widetilde{\rm N}_{n,r}$.
We denote by $\widetilde{G}_n = G_n / \langle...

For a general surface $M$ and an arbitrary braid $\alpha$ from the surface
braid group $B_{n-1}(M)$ we study the system of equations
$d_1\beta=\cdots=d_{n}\beta=\alpha, $ where operation $d_i$ is deleting of
$i$-th strand. We obtain that if $M\not=S^2$ or $\mathbb RP^2$ this system of
equations has a solution $\beta\in B_{n}(M)$ if and only if
$d_1...

We define a new operation of multiplication on the set of square matrices. We determine when this multiplication is associative and when the set of matrices with this multiplication and the ordinary addition of matrices constitutes a ring. Furthermore, we determine when the nonstandard product admits the identity element and which elements are inve...

We study twisted conjugacy classes of the unit element in different groups.
Fel'shtyn and Troitsky showed that the twisted conjugacy class of the unit
element of an abelian group is a subgroup for every automorphism. The structure
is investigated of a group whose twisted conjugacy class of the unit element is
a subgroup for every automorphism (inne...