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Viktor Lopatkin

Publications and source records attributed to Viktor Lopatkin.

At least 19 recordsLinked to original sources
arXiv2025

On finite-dimensional homogeneous Lie algebras of derivations of polynomial rings

For a finite set of homogeneous locally nilpotent derivations of the algebra of polynomials in several variables, a finite dimensionality criterion for the Lie algebra generated by these derivations is known. Also the structure of the corresponding finite-dimensional Lie algebras is described in previous works. In this paper, we obtain a finite dimensionality criterion for a Lie algebra generated by a finite set of homogeneous derivations, each of which is not locally nilpotent.

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

On meandric permutations

We give a bijection between meanders and special sorts of Gauss diagrams which aroused from the Thurston generators of braid groups. It allows us to give an algorithm to construct such diagrams and to code meanders by matrices which are exactly incident matrices of the corresponding adjacency graphs of the diagrams. Finally, we show that these matrices are idempotent over the field $\mathsf{GF}(2)$, and obtain a criterion for a permutation to be meandric.

math.AT↗
arXiv2022

Describing realizable Gauss diagrams using the concepts of parity or bipartate graphs

Two recent publications describe realizable Gauss diagrams using conditions stating that the number of chords in certain sets of chords is even or odd. We demonstrate that these descriptions are incorrect by finding multiple counter-examples. However, the idea of having a parity-based description of realizable Gauss diagrams is attractive. We recall that realizability of Gauss diagrams as touch curves can be described via bipartite graphs. We show that realizable Gauss diagrams can be described via bipartite graphs.

math.GT↗
arXiv2022

Transposed Poisson structures on Galilean and solvable Lie algebras

Transposed Poisson structures on complex Galilean type Lie algebras and superalgebras are described. It was proven that all principal Galilean Lie algebras do not have non-trivial $\frac{1}{2}$-derivations and as it follows they do not admit non-trivial transposed Poisson structures. Also, we proved that each complex finite-dimensional solvable Lie algebra admits a non-trivial transposed Poisson structure and a non-trivial ${\rm Hom}$-Lie structure.

math.RA↗
arXiv2021

$\frac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras

A relation between $\frac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras was established. Some non-trivial transposed Poisson algebras with a certain Lie algebra (Witt algebra, algebra $\mathcal{W}(a,-1)$, thin Lie algebra and solvable Lie algebra with abelian nilpotent radical) were constructed. In particular, we constructed an example of the transposed Poisson algebra with associative and Lie parts isomorphic to the Laurent polynomials and the Witt algebra. On the other side, it was proven that there are no non-trivial transposed Poisson algebras with Lie algebra part isomorphic to a semisimple finite-dimensional algebra, a simple finite-dimensional superalgebra, the Virasoro algebra, $N=1$ and $N=2$ superconformal algebras, or a semisimple finite-dimensional $n$-Lie algebra.

math.RA↗
arXiv2021

Garside Theory: a Composition--Diamond Lemma Point of View

This paper shows how to obtain the key concepts and notations of Garside theory by using the Composition--Diamond lemma. We also show in some cases the greedy normal form is exactly a Gröbner--Shirshov normal form and a family of a left-cancellative category is a Garside family, if and only if a suitable set of reductions is confluent up to some congruence on words.

math.RA↗
arXiv2021

Circle graphs (chord interlacement graphs) of Gauss diagrams: Descriptions of realizable Gauss diagrams, algorithms, enumeration

Chord diagrams, under the name of Gauss diagrams, are used in low-dimensional topology as an important tool for studying curves or knots. Those Gauss diagrams that correspond to curves or knots are called realizable. The theme of our paper is the fact that realizability of a Gauss diagram can be expressed via its circle graph. Accordingly, one can define and study realizable circle graphs (with realizability of a circle graph understood as realizability of any one of chord diagrams corresponding to the graph). Several studies contain theorems purporting to prove the fact. We check several of these descriptions experimentally and find counterexamples to the descriptions of realizable Gauss diagrams in some of these publications. We formulate new descriptions of realizable circle graphs and present an elegant algorithm for checking if a circle graph is realizable. We enumerate realizable circle graphs for small sizes and comment on these numbers. Then we concentrate on one type of curves, called meanders, and study the circle graphs of their Gauss diagrams.

math.GT↗
arXiv2020

Parafree augmented algebras and Gröbner-Shirshov bases for complete augmented algebras

We develop a theory of parafree augmented algebras similar to the theory of parafree groups and explore some questions related to the Parafree Conjecture. We provide an example of finitely generated parafree augmented algebra of infinite cohomological dimension. Motivated by this example, we prove a version of the Composition-Diamond lemma for complete augmented algebras and provide a sufficient condition for augmented algebra to be residually nilpotent on the language of its relations.

math.RA↗
arXiv2019

On Common Divisors of Fox Derivatives with towards to Zero Divisors of Group Rings

Using Composition--Diamond Lemma we construct presentations of groups $G = \langle x_1,\ldots,x_n \, | \, r_1,\ldots, r_m \rangle$ with the following property; for a fixed $1 \le i \le n$, and for all $1 \le j \le m$, Fox derivatives $\partial r_j / \partial x_i$ have common divisor. It follows that if $π_2(K) \ne 0$, where $K$ is the standard $2$-complex associated with $G$ then the group ring $\mathbb{Z}[G]$ has nontrivial zero divisors.

math.GR↗
arXiv2019

Bipartite Graphs as Polynomials, and Polynomials as Bipartite Graphs (with a view towards dividing in $\mathbb{N}[x],$ $\mathbb{N}[x,y]$)

The aim of this paper is to show that any finite undirected bipartite graph can be considered as a polynomial $p \in \mathbb{N}[x]$, and any directed finite bipartite graph can be considered as a polynomial $p\in\mathbb{N}[x,y]$, and vise verse. We also show that the multiplication in semirings $\mathbb{N}[x]$, $\mathbb{N}[x,y]$ correspondences to a operations of the corresponding graphs which looks like a ``perturbed'' products of graphs. As an application, we give a new point of view to dividing in semirings $\mathbb{N}[x]$, $\mathbb{N}[x,y]$. Finally, we endow the set of all bipartite graphs with the Zariski topology.

math.RA↗
arXiv2018

On Realizability Of Gauss Diagrams And Constructions Of Meanders

The problem of which Gauss diagram can be realized by plane curves is an old one and has been solved in several ways. In this paper, we present a direct approach to this problem. We show that needed conditions for realizability of a Gauss diagram can be interpreted as follows "the number of exits = the number of entrances" and the sufficient condition is based on Jordan curve Theorem. We give a matrix approach of realization of Gauss diagrams and then we present an algorithm to construct meanders

math.AT↗
arXiv2017

Derivations of Leavitt path algebra

In this paper, we describe the $K$-module $HH^1(L_K(Γ))$ of outer derivations of the Leavitt path algebra $L_K(Γ)$ of a row-finite graph $Γ$ with coefficients in an associative commutative ring $K$ with unit. We give an explicit formula for every outer derivation of $L_K(Γ)$. We also describe the Lie algebra structure of outer derivations of the Toeplitz algebra and we prove that every derivation of the Leavitt path algebra can be extended to a derivation of the corresponding $C^*$-algebra.

math.AT↗
arXiv2017

On Realizability of Gauss Diagrams

The problem of which Gauss diagram can be realized by knots is an old one and has been solved in several ways. In this paper, we present a direct approach to this problem. We show that the needed conditions for realizability of a Gauss diagram can be interpreted as follows "the number of exits = the number of entrances" and the sufficient condition is based on Jordan curve Theorem.

math.GT↗
arXiv2017

Petri Nets and its Polynomials

For every finite Petri net, we construct a commutative polynomial in two variables and with coefficients from the semiring of natural numbers. We also present an inverse construction and show that multiplication of polynomials correspondence to the product of the corresponding Petri nets in the category of Petri nets with Winskel's morphisms. We endow the set of all Petri nets with Zariski topology.

cs.LO↗
arXiv2016

Derivations of the Leavitt path algebra W(n)

The aim of this paper is to describe all inner and all outer derivations of Leavitt path algebra W(n) via explicit formulas. In fact the space of all inner and all outer derivations of the Leavitt path algebra W(n) has been described.

math.RA↗