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Boris Kunyavskii

Publications and source records attributed to Boris Kunyavskii.

At least 19 recordsLinked to original sources
arXiv2026

Birational properties of word varieties

We prove that the subvariety of $SL(2)\times SL(2)$ given by the matrix equation $w(X,Y)=α$, where $w$ is a word in two letters, is closely related to an explicit smooth conic bundle over the associated `trace surface' in the 3-dimensional affine space. When $w$ is the commutator word, we show that this variety can be irrational if the ground field $k$ is not algebraically closed, answering a question of Rapinchuk, Benyash-Krivetz, and Chernousov. When $k$ is a number field, it satisfies weak approximation with the Brauer--Manin obstruction.

math.AG↗
arXiv2026

Linearization of finite subgroups of Cremona groups over non-closed fields

We study linearizability properties of finite subgroups of the Cremona group ${\mathrm{Cr}}_n(k)$ in the case where $k$ is a global field, with the focus on the local-global principle. For every global field $k$ of characteristic different from 2 and every $n \ge 3$ we give an example of a birational involution of $\mathbb P^n_k$ (=an element $g$ of order $2$ in ${\mathrm{Cr}}_n(k)$) such that $g$ is not $k$-linearizable but $g$ is $k_v$-linearizable in ${\mathrm{Cr}}_n(k_v)$ for all places $v$ of $k$. The main tool is a new birational invariant generalizing those introduced by Manin and Voskresenski\uı in the arithmetic case and by Bogomolov--Prokhorov in the geometric case. We also apply it to the study of birational involutions in real plane.

math.AG↗
arXiv2024

Bracket width of current Lie algebras

The length of an element $z$ of a Lie algebra $L$ is defined as the smallest number $s$ needed to represent $z$ as a sum of $s$ brackets. The bracket width of $L$ is defined as supremum of the lengths of its elements. Given a finite-dimensional simple Lie algebra $\mathfrak g$ over an algebraically closed field $k$ of characteristic zero, we study the bracket width of current Lie algebras $L=\mathfrak g\otimes A$. We show that for an arbitrary $A$ the width is at most 2. For $A=k[[t]]$ and $A=k[t]$ we compute the width for algebras of types A and C.

math.RA↗
arXiv2024

Uniform bounded elementary generation of Chevalley groups

In this paper we establish a definitive result which almost completely closes the problem of bounded elementary generation for Chevalley groups of rank $\ge 2$ over arbitrary Dedekind rings $R$ of arithmetic type, with uniform bounds. Namely, we show that for every reduced irreducible root system $Φ$ of rank $\ge 2$ there exists a universal bound $L=L(Φ)$ such that the simply connected Chevalley groups $G(Φ,R)$ have elementary width $\le L$ for all Dedekind rings of arithmetic type $R$.

math.GR↗
arXiv2023

Tori and surfaces violating a local-to-global principle for rationality

We show that even within a class of varieties where the Brauer--Manin obstruction is the only obstruction to the local-to-global principle for the existence of rational points (Hasse principle), this obstruction, even in a stronger, base change invariant form, may be insufficient for explaining counter-examples to the local-to-global principle for rationality. We exhibit examples of toric varieties and rational surfaces over an arbitrary global field k each of those, in the absence of the Brauer obstruction to rationality, is rational over all completions of k but is not k-rational.

math.AG↗
arXiv2023

Sha-rigidity of Chevalley groups over local rings

We prove that every locally inner endomorphism of a Chevalley group (or its elementary subgroup) over a local ring with an irreducible root system of rank >1 (with 1/2 for the systems A_2, F_4, B_l, C_l and with 1/3 for the system G_2) is inner, so that all these groups are Sha-rigid.

math.GR↗
arXiv2023

Bounded generation of Steinberg groups over Dedekind rings of arithmetic type

The main result of the present paper is bounded elementary generation of the Steinberg groups $\mathrm{St}(Φ,R)$ for simply laced root systems $Φ$ of rank $\ge 2$ and arbitrary Dedekind rings of arithmetic type. Also, we prove bounded generation of $\mathrm{St}(Φ,\mathbb F_{q}[t,\,t^{-1}])$ for all root systems $Φ$, and bounded generation of $\mathrm{St}(Φ,\mathbb F_{q}[t])$ for all root systems $Φ\neq\mathsf A_1$. The proofs are based on a theorem on bounded elementary generation for the corresponding Chevalley groups, where we provide uniform bounds.

math.KT↗
arXiv2023

Tate-Shafarevich groups and algebras

The Tate-Shafarevich set of a group G defined by Takashi Ono coincides, in the case where G is finite, with the group of outer class-preserving automorphisms of G introduced by Burnside. We consider analogues of this important group-theoretic object for Lie algebras and associative algebras and establish some new structure properties thereof. We also discuss open problems and eventual generalizations to other algebraic structures.

math.GR↗
arXiv2022

The Bogomolov multiplier of finite simple groups

The subgroup of the Schur multiplier of a finite group G consisting of all cohomology classes whose restriction to any abelian subgroup of G is zero is called the Bogomolov multiplier of G. We prove that if G is quasisimple or almost simple, its Bogomolov multiplier is trivial except for the case of certain covers of PSL(3,4).

math.GR↗
arXiv2022

Bounded generation and commutator width of Chevalley groups: function case

We prove that Chevalley groups over polynomial rings $\mathbb F_q[t]$ and over Laurent polynomial $\mathbb F_q[t,t^{-1}]$ rings, where $\mathbb F_q$ is a finite field, are boundedly elementarily generated. Using this we produce explicit bounds of the commutator width of these groups. Under some additional assumptions, we prove similar results for other classes of Chevalley groups over Dedekind rings of arithmetic rings in positive characteristic. As a corollary, we produce explicit estimates for the commutator width of affine Kac--Moody groups defined over finite fields. The paper contains also a broader discussion of the bounded generation problem for groups of Lie type, some applications and a list of unsolved problems in the field.

math.GR↗
arXiv2021

Bracket width of simple Lie algebras

The notion of commutator width of a group, defined as the smallest number of commutators needed to represent each element of the derived group as their product, has been extensively studied over the past decades. In particular, in 1992 Barge and Ghys discovered the first example of a simple group of commutator width greater than one among groups of diffeomorphisms of smooth manifolds. We consider a parallel notion of bracket width of a Lie algebra and present the first examples of simple Lie algebras of bracket width greater than one. They are found among the algebras of polynomial vector fields on smooth affine varieties.

math.AG↗
arXiv2018

Geometry of word equations in simple algebraic groups over special fields

This paper contains a survey of recent developments in investigation of word equations in simple matrix groups and polynomial equations in simple (associative and Lie) matrix algebras along with some new results on the image of word maps on algebraic groups defined over special fields: complex, real, p-adic (or close to such), or finite.

math.AG↗
arXiv2018

Word maps on perfect algebraic groups

We extend Borel's theorem on the dominance of word maps from semisimple algebraic groups to some perfect groups. In another direction, we generalize Borel's theorem to some words with constants. We also consider the surjectivity problem for particular words and groups, give a brief survey of recent results, present some generalizations and variations and discuss various approaches, with emphasis on new ideas, constructions and connections.

math.GR↗
arXiv2018

Word maps, word maps with constants and representation varieties of one-relator groups

We consider word maps and word maps with constants on a simple algebraic group. We present results on the images of such maps, in particular, we prove a theorem on the dominance of general word maps with constants, which can be viewed as an analogue of a well-known theorem of Borel on the dominance of genuine word maps. Besides, we establish a relationship between the existence of unipotents in the image of a word map and the structure of the corresponding representation variety.

math.GR↗
arXiv2015

Word maps in Kac-Moody setting

The paper is a short survey of recent developments in the area of word maps evaluated on groups and algebras. It is aimed to pose questions relevant to Kac--Moody theory.

math.GR↗
arXiv2014

Stably Cayley semisimple groups

A linear algebraic group G over a field k is called a Cayley group if it admits a Cayley map, i.e. a G-equivariant birational isomorphism over k between the group variety G and its Lie algebra Lie(G). A prototypical example is the classical "Cayley transform" for the special orthogonal group SO(n) defined by Arthur Cayley in 1846. A linear algebraic k-group G is called stably Cayley if $G \times S$ is Cayley for some split k-torus S. We classify stably Cayley semisimple groups over an arbitrary field k of characteristic 0.

math.AG↗