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Vsevolod Gubarev

Publications and source records attributed to Vsevolod Gubarev.

At least 37 records · Page 2Linked to original sources
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.

math.GR↗
arXiv2022

Spectrum of Rota-Baxter operators

We prove that the spectrum of every Rota-Baxter operator of weight $λ$ on a unital algebraic (not necessarily associative) algebra over a field of characteristic zero is a subset of $\{0,-λ\}$. For a finite-dimensional unital algebra the same statement is shown to hold without a restriction on the characteristic of the ground field. Based on these results, we define the Rota-Baxter $λ$-index $\mathrm{rb}_λ(A)$ of an algebra $A$ as the infimum of the degrees of minimal polynomials of all Rota-Baxter operators of weight $λ$ on $A$. We calculate the Rota-Baxter $λ$-index for the matrix algebra $M_n(F)$, $\mathrm{char}\,F = 0$: it is shown that $\mathrm{rb}_λ(M_n(F)) = 2n-1$.

math.RA↗
arXiv2022

An example of a simple double Lie algebra

We extend the correspondence between double Lie algebras and skew-symmetric Rota-Baxter operators of weight 0 on the matrix algebra for the infinite-dimensional case. We give the first example of a simple double Lie algebra.

math.RA↗
arXiv2021

Rota-Baxter operators on groups

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. We state the connection between Rota-Baxter operators on a group and Rota-Baxter operators on an associated Lie ring. We describe Rota-Baxter operators on sporadic simple groups.

math.GR↗
arXiv2020

Injective Rota-Baxter operators of weight zero on $F[x]$

Rota-Baxter operators present a natural generalisation of integration by parts formula for the integral operator. In 2015, Zheng, Guo, and Rosenkranz conjectured that every injective Rota-Baxter operator of weight zero on the polynomial algebra $\mathbb{R}[x]$ is a composition of the multiplication by a nonzero polynomial and a formal integration at some point. We confirm this conjecture over any field of characteristic zero. Moreover, we establish a structure of an ind-variety on the moduli space of these operators and describe an additive structure of generic modality two on it. Finally, we provide an infinitely transitive action on codimension one subsets.

math.RA↗
arXiv2020

Monomial Rota-Baxter operators on free commutative non-unital algebra

A Rota-Baxter operator defined on the polynomial algebra is called monomial if it maps each monomial to a monomial with some coefficient. We classify monomial Rota-Baxter operators defined on the algebra of polynomials in one variable without constant term. We also describe injective monomial Rota--Baxter operators of nonzero weight on the algebra of polynomials in several variables without constant term.

math.RA↗
arXiv2019

Rota-Baxter operators on unital algebras

We state that all Rota-Baxter operators of nonzero weight on Grassmann algebra over a field of characteristic zero are projections on a subalgebra along another one. We show the one-to-one correspondence between the solutions of associative Yang-Baxter equation and Rota-Baxter operators of weight zero on the matrix algebra $M_n(F)$ (joint with P. Kolesnikov). We prove that all Rota-Baxter operators of weight zero on a unital associative (alternative, Jordan) algebraic algebra over a field of characteristic zero are nilpotent. For an algebra $A$, we introduce its new invariant the rb-index $\mathrm{rb}(A)$ as the nilpotency index for Rota-Baxter operators of weight zero on $A$. We show that $\mathrm{rb}(M_n(F)) = 2n-1$ provided that characteristic of $F$ is zero.

math.RA↗
arXiv2019

In memory of Igor Dmitrievich Ado

We give a translation from Russian into English of the article "In memory of Igor Dmitrievich Ado" written by A.V. Dorodnov and I.I. Sakhaev and published in Izv.\ Vyssh.\ Uchebn.\ Zaved.\ Mat.\ no. 8, (1984), 87--88. It is an orbituary for I. D. Ado. A translation might be useful in general, and in particular for a possible Wikipedia entry of Ado's life in English. In the references we list all known $12$ publications of I.D. Ado, taken from the article and the MATHSCINET of the AMS. The original orbituary only lists $9$ publications.

math.HO↗
arXiv2019

Rota--Baxter operators and post-Lie algebra structures on semisimple Lie algebras

Rota--Baxter operators $R$ of weight $1$ on $\mathfrak{n}$ are in bijective correspondence to post-Lie algebra structures on pairs $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{n}$ is complete. We use such Rota--Baxter operators to study the existence and classification of post-Lie algebra structures on pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{n}$ is semisimple. We show that for semisimple $\mathfrak{g}$ and $\mathfrak{n}$, with $\mathfrak{g}$ or $\mathfrak{n}$ simple, the existence of a post-Lie algebra structure on such a pair $(\mathfrak{g},\mathfrak{n})$ implies that $\mathfrak{g}$ and $\mathfrak{n}$ are isomorphic, and hence both simple. If $\mathfrak{n}$ is semisimple, but $\mathfrak{g}$ is not, it becomes much harder to classify post-Lie algebra structures on $(\mathfrak{g},\mathfrak{n})$, or even to determine the Lie algebras $\mathfrak{g}$ which can arise. Here only the case $\mathfrak{n}=\mathfrak{sl}_2(\Bbb{C})$ was studied. In this paper we determine all Lie algebras $\mathfrak{g}$ such that there exists a post-Lie algebra structure on $(\mathfrak{g},\mathfrak{n})$ with $\mathfrak{n}=\mathfrak{sl}_2(\Bbb{C})\oplus \mathfrak{sl}_2(\Bbb{C})$.

math.RA↗
arXiv2019

Decompositions of algebras and post-associative algebra structures

We introduce post-associative algebra structures and study their relationship to post-Lie algebra structures, Rota--Baxter operators and decompositions of associative algebras and Lie algebras. We show several results on the existence of such structures. In particular we prove that there exists no post-Lie algebra structure on a pair $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{n}$ is a simple Lie algebra and $\mathfrak{g}$ is a reductive Lie algebra, which is not isomorphic to $\mathfrak{n}$. We also show that there is no post-associative algebra structure on a pair $(A,B)$ arising from a Rota--Baxter operator of $B$, where $A$ is a semisimple associative algebra and $B$ is not semisimple. The proofs use results on Rota--Baxter operators and decompositions of algebras.

math.RA↗
arXiv2019

Poincare-Birkhoff-Witt theorem for pre-Lie and postLie algebras

We construct the universal enveloping preassociative and postassociative algebra for a pre-Lie and a postLie algebra respectively. We show that the pairs $(\mathrm{preLie},\mathrm{preAs})$ and $(\mathrm{postLie},\mathrm{postAs})$ are Poincare-Birkhoff-Witt-pairs, for the first one it's a reproof of the result of V. Dotsenko and P. Tamaroff.

math.RA↗
arXiv2018

Rota-Baxter operators on a sum of fields

We count the number of all Rota-Baxter operators on a finite direct sum $A = F\oplus F\oplus \ldots \oplus F$ of fields and count all of them up to conjugation with an automorphism. We also study Rota-Baxter operators on $A$ corresponding to a decomposition of $A$ into a direct vector space sum of two subalgebras. We show that every algebra structure induced on $A$ by a Rota-Baxter of nonzero weight is isomorphic to $A$.

math.RA↗
arXiv2018

Rota-Baxter operators and Bernoulli polynomials

We develop the connection between Rota-Baxter operators arisen from algebra and mathematical physics and Bernoulli polynomials. We state that a trivial property of Rota-Baxter operators implies the symmetry of the power sum polynomials and Bernoulli polynomials. We show how Rota-Baxter operators equalities rewritten in terms of Bernoulli polynomials generate identities for the last.

math.NT↗
arXiv2018

$PC$-polynomial of graph

We define $PC$-polynomial of graph which is related to clique, (in)dependence and matching polynomials. The growth rate of partially commutative monoid is equal to the largest root $β(G)$ of $PC$-polynomial of the corresponding graph. The random algebra is defined in such way that its growth rate equals the largest root of $PC$-polynomial of random graph. We prove that for almost all graphs all sufficiently large real roots of $PC$-polynomial lie in neighbourhoods of roots of $PC$-polynomial of random graph. We show how to calculate the series expansions of the latter roots. The average value of $β(G)$ over all graphs with the same number of vertices is computed. We found the graphs on which the maximal value of $β(G)$ with fixed numbers of vertices and edges is reached. From this, we derive the upper bound of $β(G)$. Modulo one assumption, we do the same for minimal value of $β(G)$. We study the Nordhaus---Gaddum bounds of $β(G)+β(\bar{G})$ and $β(G)β(\bar{G})$.

math.CO↗