Vladimir L. Popov

Vladimir L. Popov
Russian Academy of Sciences | RAS · Steklov Mathematical Institute

Prof. Dr.

About

254
Publications
27,100
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
2,061
Citations
Additional affiliations
January 2002 - present
Russian Academy of Sciences
Position
  • Fellow

Publications

Publications (254)
Article
Full-text available
We prove that the singular locus of the commuting variety of a noncommutative reductive Lie algebra is contained in the irregular locus and we compute the codimension of the latter. We prove that one of the irreducible components of the irregular locus has codimension 4. This yields the lower bound of the codimension of the singular locus, in parti...
Article
Full-text available
We show that if a field k contains sufficiently many elements(for instance, if k is infinite), and K is an algebraically closed field containing k, then every linear algebraic k-group over K is k-isomorphic to Aut(A\otimes_kK), where A is a finite dimensional simple algebra over k.
Article
Full-text available
The classical Cayley map, X --> (I_n-X)/(I_n+X), is a birational isomorphism between the special orthogonal group SO_n and its Lie algebra so_n, which is SO_n-equivariant with respect to the conjugating and adjoint actions respectively. We ask whether or not maps with these properties can be constructed for other algebraic groups. We show that the...
Article
Full-text available
For the coordinate algebras of connected affine algebraic groups, we explore the problem of finding a presentation by generators and relations canonically determined by the group structure.
Book
Full-text available
These are the notes recording the contents of a course of five lectures given by the author at the Mathematical Institute of the Reijksuniversiteit Utrecht in October 1980. The aim of these lectures was to develop the theory of discrete groups generated by affine unitary reflections, and in fact, to provide a classification of these groups. This th...
Presentation
Full-text available
Abstract. There are many examples where the problem of classifying alge- braic objects of a certain type is reformulated as that of classifying orbits of some algebraic group action. The talk is aimed to discuss the decidability of the equivalence problem for two objects of the considered type in such cases.
Preprint
Full-text available
A new infinite series of rational affine algebraic varieties is constructed whose automorphism group contains the automorphism group ${\rm Aut}(F_n)$ of the free group $F_n$ of rank $n$. The automorphism groups of such varieties are nonlinear and contain the braid group $B_n$ on $n$ strands for $n\geqslant 3$, and are nonamenable for $n\geqslant 2$...
Preprint
Full-text available
Considering a certain construction of algebraic varieties $X$ endowed with an algebraic action of the group ${\rm Aut}(F_n)$, $n<\infty$, we obtain a criterion for the faithfulness of this action. It gives an infinite family $\mathscr F$ of $X$'s such that ${\rm Aut}(F_n)$ embeds into ${\rm Aut}(X)$. For $n\geqslant 3$, this implies nonlinearity, a...
Preprint
Full-text available
We prove that every orbit of the adjoint representation of any connected reductive algebraic group $G$ is a rational algebraic variety. For complex simply connected semisimple $G$, this implies rationality of affine Hamiltonian $G$-varieties (which we classify).
Article
We explore to what extent the group variety of an algebraic group determines its group structure.
Article
Full-text available
We explore whether a root lattice may be similar to the lattice O of integers of a number field K endowed with the inner product (x, y) := TraceK/Q(x · θ(y)), where θ is an involution of K. We classify all pairs K, θ such that O is similar to either an even root lattice or the root lattice Z^[K:Q]. We also classify all pairs K, θ such that O is a r...
Article
В работе исследуется, в какой мере групповое многообразие алгебраической группы определяет ее групповую структуру.
Presentation
Full-text available
The talk is aimed to discuss to what extent the group variety of a connected algebraic group or the group manifold of a connected real Lie group determines its group structure.
Presentation
Full-text available
Лекции, прочитанные 21, 22 и 23 августа 2021 г в Самаре на школе-конференции "Алгебры Ли, алгебраические группы и теория инвариантов". Цель лекций---описать классификацию дискретных групп, порожденных комплексными отражениями эрмитовых аффинных пространств. [Lectures (in Russian) delivered on August 21, 22 and 23, 2021 in Samara at the school-conf...
Preprint
Full-text available
For every positive integer $n$, we construct, using algebraic groups, an infinite family of irreducible algebraic varieties $X$, whose automorphism group ${\rm Aut}(X)$ contains the automorphism group ${\rm Aut}(F_n)$ of a free group $F_n$ of rank $n$ as a subgroup. This property implies that, for $n \geqslant 2$, such groups ${\rm Aut}(X)$ are non...
Article
Full-text available
We explore connected affine algebraic groups G, which enjoy the following finiteness property (F): for every algebraic action of G, the closure of every G-orbit contains only finitely many G-orbits. We obtain two main results. First, we classify such groups. Namely, we prove that a connected affine algebraic group G enjoys property (F) if and only...
Preprint
Full-text available
We explore to what extent the underlying variety of a connected algebraic group or the underlying manifold of a real Lie group determines its group structure.
Presentation
Full-text available
The following construction of a root system of type G2 is known (see J.-P. Serre’s book “Lie Algebras and Lie Groups”, page 315 of the Russian translation): “It can be described as the set of algebraic integers of a cyclotomic field generated by a cubic root of unity, with norm 1 and 3”. The talk, based on a joint work with Yu. G. Zarhin, concerns...
Article
We classify the types of root systems R in the rings of integers of number fields K such that the Weyl group W (R) lies in the group L(K) generated by Aut(K) and multiplications by the elements of K *. We also classify the Weyl groups of root systems of rank n which are isomorphic to a subgroup of L(K) for a number field K of degree n over Q.
Preprint
Full-text available
Several results on presenting an affine algebraic group variety as a product of algebraic varieties are obtained.
Presentation
Full-text available
The following construction of the root system of type G_2 is known (see the book J.-P. Serre "Lie algebras and Lie groups", p. 315 of the Russian translation): “It can be described as a set algebraic integers of the cyclotomic field generated by the cube root of 1, with the norm 1 or 3". This talk, based on the joint results with Yu. G. Zarhin, i...
Article
Full-text available
We explore whether a root lattice may be similar to the lattice O of integers of a number field K endowed with the inner product (x, y) := TraceK/Q(x · θ(y)), where θ is an involution of K. We classify all pairs K, θ such that O is similar to either an even root lattice or the root lattice Z^[K:Q]. We also classify all pairs K, θ such that O is a r...
Article
This paper investigates whether a root lattice can be similar to a lattice O of integer elements of a number field K endowed with the inner product (x,y):=Trace_{K/\mathbb Q}(x\cdot\theta(y)) , where \theta is an involution of the field K. For each of the following three properties (1), (2), (3), a classification of all the pairs K, \theta with thi...
Article
В работе исследуется, может ли корневая решетка быть подобна решетке O всех целых элементов числового поля K, снабженной внутренним произведением (x,y):= Trace_{K/Q}(x.\theta(y)), где \theta ----инволюция поля K. Для каждого из следующих трех свойств (1), (2), (3) получена классификация всех пар K, \theta, обладающих этим свойством: (1) O является...
Data
In the first version of this paper (arXiv:1707.06914v1), the main statement is proved under the restriction that the characteristic of the ground field is equal to zero. In the current version this statement is proved for arbitrary characteristic.
Chapter
Full-text available
This is an expanded version of the talk by the author at the conference Polynomial Rings and Affine Algebraic Geometry, February 12–16, 2018, Tokyo Metropolitan University, Tokyo, Japan. Considering a local version of the Zariski Cancellation Problem naturally leads to exploration of some classes of varieties of special kind and their equivariant v...
Article
A brief biography of Igor Rostislavovich Shafarevich and an overview of all his main scientific publications are presented.
Article
Full-text available
Изложена краткая биография Игоря Ростиславовича Шафаревича, и приведен обзор всех его основных опубликованных научных результатов.
Preprint
Full-text available
We explore whether a root lattice may be similar to the lattice $\mathscr O$ of integers of a number field $K$ endowed with the inner product $(x, y):={\rm Trace}_{K/\mathbb Q}(x\cdot\theta(y))$, where $\theta$ is an involution of $K$. We classify all pairs $K$, $\theta$ such that $\mathscr O$ is similar to either an even root lattice or the root l...
Article
В работе доказано, что для всякого простого числа p существуют алгебраическое действие двумерной группы Витта W_2(p) на алгебраическом многообразии X и точка x ∈ X такие, что замыкание W_2(p)-орбиты точки x в X содержит бесконечно много W_2(p)-орбит. Это связано с задачей о распространении со случая нулевой характеристики на случай характеристики p...
Article
We prove that for any prime p there exists an algebraic action of the two-dimensional Witt group W2 (p) on an algebraic variety X such that the closure in X of the W2(p)-orbit of some point x ∈ X contains infinitely many W2(p)-orbits. This is related to the problem of extending, from the case of characteristic zero to the case of characteristic p,...
Book
Full-text available
Том посвящен памяти выдающегося математика, академика Игоря Ростиславовича Шафаревича. В нем представлены работы по алгебре, теории чисел и алгебраической геометрии, авторами которых являются представители математической школы, созданной И.Р. Шафаревичем.
Article
Full-text available
Изложена краткая биография Игоря Ростиславовича Шафаревича, и приведен обзор всех его основных опубликованных научных результатов.
Article
Full-text available
Мы доказываем, что для всякого целого простого числа $p\geqslant 2$ существуют алгебраическое действие двумерной группы Витта $W_2(p)$ на алгебраическом многообразии $X$ и точка $x\in X$ такие, что замыкание $W_2(p)$-орбиты точки $x$ в $X$ содержит бесконечно много $W_2(p)$-орбит. Это связано с задачей о распространении с нулевой характеристики на...
Presentation
Full-text available
I shall consider algebraic group actions transitive on general tuples of points andexplain a relation of this topic to the existence problem of special series of tensor product of simple modules. I shall also discuss the relevant classification problems and an open question on the root systems.
Presentation
Full-text available
The results obtained by the author jointly with Yu. G. Zarhin on the realization of root systems in number fields and on the realization of lattices similar to the root ones as lattices of all integer elements of number fields equipped with a twisted trace form are presented.
Article
The first group of results of the paper concerns the compressibility of finite subgroups of the Cremona groups. The second concerns the embeddability of other groups in the Cremona groups and, conversely, of the Cremona groups in other groups. The third concerns the connectedness of the Cremona groups.
Article
Let Y be a smooth irreducible projective algebraic variety which is birationally isomorphic to the variety of flexes of plane cubics. We prove that, for any positive integer d, there are no non-zero regular differential d-forms on Y. For d=1 it is the main result of the paper V. S. Kulikov, "On the variety of the inflection points of plane cubic c...
Article
Первая группа результатов этой работы касается сжимаемости конечных подгрупп групп Кремоны. Вторая – вложимости других групп в группы Кремоны и, наоборот, групп Кремоны в другие группы. Третья – связности групп Кремоны.
Presentation
Full-text available
Video recording of the talk (in Russian) ``Jordan groups''; available at https://www.youtube.com/watch?v=k49VJ4sFtps&list=PLO5xfXeseNJ4kdKeBRractfCv-h0Haw_M&index=17&t=0s
Article
В работе доказано, что на каждой десингуляризации многообразия точек перегиба плоских кубик нет ненулевых регулярных дифференциальных форм любой степени d. Для d=1 это является основным результатом работы S. Kulikov, On the variety of the inflection points of plane cubic curves, 2018, 27 pp., arXiv:1810.01705v1
Presentation
ABSTRACT. In many fields, from theoretical mechanics to number theory, various canonical forms of transformations play a significant role. It follows from the theory of Jordan normal form that any one-parameter group of affine transformations of n-dimensional coordinate space is reduced by an affine transformation of coordinates to a triangular for...
Article
Доказано, что основной результат работы [1] является частным случаем более общего утверждения, которое можно вывести с помощью короткого рас- суждения из классических теорем Ричардсона и Луны.
Article
Full-text available
It is shown that the main result of N. R. Wallach, Principal orbit type theorems for reductive algebraic group actions and the Kempf-Ness Theorem, arXiv:1811.07195v1 (17 Nov 2018), is a special case of a more general statement, which can be deduced, using a short argument, from the classical Richardson and Luna theorems.
Preprint
It is shown that the main result of N. R. Wallach, Principal orbit type theorems for reductive algebraic group actions and the Kempf--Ness Theorem, arXiv:1811.07195v1 (17 Nov 2018) is a special case of a more general statement, which can be deduced, using a short argument, from the classical Richardson and Luna theorems.
Preprint
Full-text available
It is shown that the main result of N. R. Wallach, Principal orbit type theorems for reductive algebraic group actions and the Kempf-Ness Theorem , arXiv:1811.07195v1 (17 Nov 2018) is a special case of a more general statement, which can be deduced, using a short argument, from the classical Richardson and Luna theorems.
Preprint
Full-text available
This is an expanded version of the talk by the author at the conference Polynomial Rings and Affine Algebraic Geometry, February 12--16, 2018, Tokyo Metropolitan University, Tokyo, Japan. Considering a local version of the Zariski Cancellation Problem naturally leads to exploration of some classes of varieties of special kind and their equi-variant...
Article
Первая группа результатов этой работы касается сжимаемости конечных подгрупп групп Кремоны. Вторая - вложимости других групп в группы Кремоны и, наоборот, групп Кремоны в другие группы. Третья - связности групп Кремоны. Библиография: 41 наименование.
Presentation
Full-text available
ABSTRACT. So far, in the studies of finite subgroups of Cremona groups, they were all considered on an equal footing. However, in reality it is necessary to consider some of them as “not basic”, since they are obtained from others by a standard “base change” construction. This leads to the problem of finding the subgroups in the classification list...
Book
Full-text available
This volume, dedicated to the memory of the great American mathematician Bertram Kostant (May 24, 1928 – February 2, 2017), is a collection of 19 invited papers by leading mathematicians working in Lie theory, representation theory, algebra, geometry, and mathematical physics. Kostant’s fundamental work in all of these areas has provided deep new i...
Preprint
Full-text available
We classify the types of root systems R in the rings of integers of number fields K such that the Weyl group W (R) lies in the group L(K) generated by Aut(K) and multiplications by the elements of K *. We also classify the Weyl groups of roots systems of rank n which are isomorphic to a subgroup of L(K) for a number field K of degree n over Q.
Preprint
Full-text available
The first group of results of this paper concerns the compressibility of finite subgroups of the Cremona groups. The second concerns the embeddability of other groups in the Cremona groups and, conversely, the Cremona groups in other groups. The third concerns the connectedness of the Cremona groups.
Preprint
Full-text available
The first group of results of this paper concerns the compressibility of finite subgroups of the Cremona groups. The second concerns the embeddability of other groups in the Cremona groups and, conversely, the Cremona groups in other groups. The third concerns the connectedness of the Cremona groups.
Article
The compressibility of certain types of finite groups of birational automorphisms of algebraic varieties is established.
Article
Установлена сжимаемость некоторых типов конечных групп бирациональных автоморфизмов алгебраических многообразий.
Article
Full-text available
We classify the types of root systems R in the rings of integers of number elds K such that the Weyl group W (R) lies in the group generated by Aut(K) and multiplications by the elements of K *.
Article
Мы классифицируем типы таких систем корней R в кольцах целых числовых полей K, что группа Вейля W(R) лежит в порожденной Aut(K) и умножениями из K группе.
Preprint
Full-text available
We classify the types of root systems R in the rings of integers of number fields K such that the Weyl group W (R) lies in the group L(K) generated by Aut(K) and multiplications by the elements of K *. We also classify the Weyl groups of roots systems of rank n which are isomorphic to a subgroup of L(K) for a number field K of degree n over Q.
Article
We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This implies that all algebraic groups (not necessarily affine) over fields of characteristic zero and some transformation groups of complex spaces and Riemannian manifods are Jordan.
Preprint
Full-text available
We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This implies that all algebraic groups (not necessarily affine) over fields of characteristic zero and some transformation groups of complex spaces and Riemannian manifolds are Jordan.
Chapter
Full-text available
We first establish several general properties of modality of algebraic group actions. In particular,we introduce the notion of a modality-regular action and prove that every visible action is modality-regular. Then, using these results, we classify irreducible linear representations of connected simple algebraic groups of every fixed modality ≤ 2....
Conference Paper
Full-text available
It is an abstract of a minicourse (3 lectures) on discrete groups generated by complex reflections delievered at the VIth Conference on Algebraic Geometry and Complex Analysis for Young Mathematicians of Russia (August 25–30, 2017, Nothern (Arctic) Federal University named after M. V. Lomonosov, Koryazhma, Arkhangelsk region, Russia).
Article
We prove that the affine-triangular subgroups are Borel subgroups of Cremona groups
Article
We first establish several general properties of modality of algebraic group actions. In particular, we introduce the notion of a modality-regular action and prove that every visible action is modality-regular. Then, using these results, we classify irreducible linear representations of connected simple algebraic groups of every fixed modality $\le...
Chapter
Full-text available
Exploring Bass' Triangulability Problem on unipotent algebraic subgroups of the affine Cremona groups, we prove a triangulability criterion, the existence of nontriangulable connected solvable affine algebraic subgroups of the Cremona groups, and stable triangulability of such subgroups; in particular, in the stable range we answer Bass' Triangulab...
Article
Full-text available
Мы доказываем, что аффинно-треугольные подгруппы являются борелевс- кими подгруппами групп Кремоны.
Article
Пусть G – связная линейная алгебраическая группа, определённая над алгебраически зам- кнутым полем k характеристики нуль, V – конечномерное векторное пространство над k и ρ: G → GL(V) – алгебраическое представление. Модальностью представления ρ называется модаль- ность определённого им действия группы G на V. В работе даны классификации всех предст...
Presentation
Full-text available
The talk is aimed to elaborate on the Bass' Triangulability Problem concerning rational triangulability of unipotent algebraic subgroups of the affine Cremona groups. The following topics will be discussed: a rational triangulability criterion; the existence of rationally non-triangulable connected solvable affine algebraic subgroups of the Cremon...
Conference Paper
Full-text available
The following theorems are announced: (1) Every orbit of the adjoint representation of any connected reductive algebraic group over an algebraically closed field of characteristic zero is a rational algebraic variety. (2) Every simply connected symplectic algebraic variety homogeneous with respect to a connected complex reductive algebraic group is...
Data
Full-text available
For the coordinate algebras of abelian varieties, the problem of finding a presentation by generators and relations canonically determined by the group structure has been explored and solved by D. Mumford. Since every connected algebraic group is an extension of a connected linear algebraic group by an abelian variety, the analogous problem for con...
Article
A general theorem on the purity of invariant field extensions is proved. Using it, a criterion of rational triangulability of connected solvable affine algebraic subgroups of the Cremona groups is obtained. This criterion is applied for proving the existence of rationally nontriangulable subgroups of the above form and for proving their stable rati...
Article
Full-text available
For every algebraically closed field k of characteristic different from 2, we prove the following: (1) Finite-dimensional (not necessarily associative) k-algebras of general type of a fixed dimension, considered up to isomorphism, are parametrized by the values of a tuple of algebraically independent (over k) rational functions of the structure con...
Article
Для любого алгебраически замкнутого поля k характеристики, отличной от 2, мы доказываем следующее: (1) конечномерные (не обязательно ассоциативные) k-алгебры общего типа фиксированной размерности, рассматриваемые с точностью до изоморфизма, параметризуются значениями некоторой последовательности алгебраически независимых над k рациональных функций...
Conference Paper
Full-text available
This an abstract of the talk intended for the Workshop on Reflection Groups at SISSA (Trieste, Italy), January 13–21, 1998.
Article
Full-text available
According to the classical theorem, every irreducible algebraic variety endowed with a nontrivial rational action of a connected linear algebraic group is birationally isomorphic to a product of another algebraic variety and P s with positive s. We show that the classical proof of this theorem actually works only in characteristic 0 and we give a c...

Network

Cited By

Projects