arXiv subjects

Mikhail V. Neshchadim

Publications and source records attributed to Mikhail V. Neshchadim.

14 recordsLinked to original sources

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.

math.RA↗

arXiv2022

On local sharply n-transitive groups

The paper is devoted to generalizations of actions of topological groups on manifolds. Instead of a topological group, we consider a local topological group generalizing the notion of a~germ or a~neighborhood in a topological group. The notion of an action of a local group on a topological space is introduced. The paper constructs the theory of local sharply $n$-transitive groups and local $n$-pseudofields. Local sharply $n$-transitive groups are reduced to simpler algebraic objects -- local $n$-pseudofields, similarly to the way Lie groups are reduced to Lie algebras, and sharply two-transitive groups, are reduced to neardomains. This can be useful, since, opposite to locally compact and connected sharply $n$-transitive groups, which are absent for $n > 3$, local sharply $n$-transitive groups exist for any $n$, for example, the group $GL_n(\mathbb{R})$. Being boundedly sharply $n$-transitive, the groups under consideration are also Lie groups, which gives extra methods for their study.

math.GR↗

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.

math.GT↗

arXiv2020

On the lower central series of some virtual knot groups

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 of knot groups. Also, we study decomposition of virtual knot groups as semi direct product and free product with amalgamation. In particular, we prove that the groups of some virtual knots are extensions of finitely generated free groups by infinite cyclic groups.

math.GT↗

arXiv2020

Exterior and symmetric (co)homology of groups

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

math.GR↗

arXiv2020

On $λ$-homomorphic skew braces

For a skew left brace $(G, \cdot, \circ)$, the map $λ: (G, \circ) \to \mathrm{Aut} \;(G, \cdot),~~a \mapsto λ_a$, 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 $\mathrm{Aut}\; (G, \cdot)$, which, in general, may not be a homomorphism. We study skew left braces $(G, \cdot, \circ)$ for which $λ: (G, \cdot) \to \mathrm{Aut}\; (G, \cdot)$ is a homomorphism. Such skew left braces will be called $λ$-homomorphic. We formulate necessary and sufficient conditions under which a given homomorphism $λ: (G, \cdot) \to \mathrm{Aut}\; (G, \cdot)$ gives rise to a skew left brace, which, indeed, is $λ$-homomorphic. As an application, we construct skew left braces when $(G, \cdot)$ is either a free group or a free abelian group. We prove that any $λ$-homomorphic skew left brace is an extension of a trivial skew brace by a trivial skew brace. Special emphasis is given on $λ$-homomorphic skew left brace for which the image of $λ$ is cyclic. A complete characterization of such skew left braces on the free abelian group of rank two is obtained.

math.RA↗

arXiv2019

Computing skew left braces of small orders

We improve Algorithm 5.1 of [Math. Comp. {\bf 86} (2017), 2519-2534] for computing all non-isomorphic 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.

math.RA↗

arXiv2018

Linearity problem for non-abelian tensor products

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. We prove that the Peiffer square of finitely generated linear groups is linear. At the end we construct faithful linear representations for the non-abelian tensor square of free group and free nilpotent group.

math.GR↗

arXiv2018

Commutator Subgroups of Virtual and Welded Braid Groups

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 $n \geq 4$, and $WB_n'$ is finitely generated for $n \geq 3$. Also we prove that $VB_3'/VB_3'' =\mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus\mathbb{Z}_3 \oplus \mathbb{Z}^{\infty}$, $VB_4' / VB_4'' = \mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_3$, $WB_3'/WB_3'' = \mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus\mathbb{Z}_3 \oplus \mathbb{Z},$ $WB_4'/WB_4'' = \mathbb{Z}_3,$ and for $n \geq 5$ the commutator subgroups $VB_n'$ and $WB_n'$ are perfect, i.e. the commutator subgroup is equal to the second commutator subgroup.

math.GT↗

arXiv2017

Automorphisms of pure braid Groups

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 and extension problem for automorphisms of some well-known exact sequences arising from braid groups, and prove that that answers are negative in most cases. Specifically, we prove that no non-trivial central automorphism of $P_n$ can be extended to an automorphism of $B_n$.

math.GR↗

arXiv2015

Palindromic Automorphisms of Free Nilpotent Groups

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 3. In particular, we obtain a generating set for the group of central palindromic automorphisms of these groups. In the end, we determine central palindromic automorphisms of free nilpotent groups of step 3 which satisfy the necessary condition of Bryant-Gupta-Levin-Mochizuki for a central automorphism to be tame.

math.GR↗

arXiv2015

Example of non-linearizable quasi-cyclic subgroup of automorphism group of polynomial algebra

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 over the field of complex numbers, which is not conjugated with a subgroup of linear automorphisms.

math.GR↗

arXiv2014

Upper central series for the group of unitriangular automorphisms of a free associative algebra

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 $U_n$ are non-linear for all $n \geq 2$.

math.GR↗

arXiv2010

Groups of triangular automorphisms of a free associative algebra and a polynomial algebra

We study a structure of the group of unitriangular automorphisms of a free associative algebra and a polynomial algebra and prove that this group is a semi direct product of abelian groups. Using this decomposition we describe a structure of the lower central series and the series of derived subgroups for the group of unitriangular automorphisms and prove that every element from the derived subgroup is a commutator. In addition we prove that the group of unitriangular automorphisms of a free associative algebra of rang more than 2 is not linear and describe some two-generated subgroups from these group. Also we give a more simple system of generators for the group of tame automorphisms than the system from Umirbaev's paper.

math.GR↗