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## The Jordan property for Lie groups and automorphism groups of complex spaces

Cornell University
,
2018.
No. 1804.00323v1.

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 cha\-racte\-ristic zero and some transformation groups of complex spaces and Riemannian manifods are Jordan.

Zhukova N. I., Mathematical notes 2013 Vol. 93 No. 5-6 P. 928–931

In this paper a unified method for studying foliations with transversal parabolic geometry of rank one is presented.
Ideas of Fraces' paper on parabolic geometry of rank one and of works of the author on conformal foliations
are developed. ...

Added: October 19, 2014

Р.С. Авдеев, Горфинкель Н. Е., Функциональный анализ и его приложения 2012 Т. 46 № 3 С. 1–15

For all spherical homogeneous spaces G/H, where G is a simply connected semisimple algebraic group and H a connected solvable subgroup of G, we compute the spectra of representations of G on spaces of regular sections of homogeneous line bundles over G/H. ...

Added: February 25, 2014

Shirokov D., Advances in Applied Clifford Algebras 2010 Vol. 20 No. 2 P. 411–425

In this paper we present new formulas, which represent commutators and anticommutators of Clifford algebra elements as sums of elements of different ranks. Using these formulas we consider subalgebras of Lie algebras of pseudo-unitary groups. Our main techniques are Clifford algebras. We have found 12 types of subalgebras of Lie algebras of pseudo-unitary groups. ...

Added: June 16, 2015

Shirokov D., Advances in Applied Clifford Algebras 2012 Vol. 22 No. 1 P. 243–256

We present a new classification of Clifford algebra elements. Our classification is based on the notion of quaternion type. Using this classification we develop a method for analyzing commutators and anticommutators of Clifford algebra elements. This method allows us to find out and prove a number of new properties of Clifford algebra elements. ...

Added: June 16, 2015

Zhukova N.I., K. I. Sheina, / Cornell University. Series math "arxiv.org". 2015. No. 1410.1144.

We get sufficient conditions for the full basic automorphism group of a complete
Cartan foliation to admit a unique (finite-dimensional) Lie group structure in the category
of Cartan foliations. In particular, we obtain sufficient conditions for this group
to be discrete. Emphasize that the transverse Cartan geometry may be noneffective.
Some estimates of the dimension of this group depending ...

Added: November 10, 2014

K. I. Sheina, N. I. Zhukova, Lobachevskii Journal of Mathematics 2018 Vol. 39 No. 2 P. 271–280

For a complete Cartan foliation (M; F) we introduce
two algebraic invariants g0(M; F) and g1(M; F) which we call structure
Lie algebras. If the transverse Cartan geometry of (M; F) is eective
then g0(M; F) = g1(M; F). We prove that if g0(M; F) is zero then in
the category of Cartan foliations the group of all basic ...

Added: March 23, 2017

Kharchev S. M., Khoroshkin S. M., Advances in Mathematics 2020 Vol. 375 No. 107368 P. 1–56

We obtain certain Mellin-Barnes integrals which present Whittaker wave functions related to classical real split forms of simple complex Lie groups ...

Added: October 18, 2020

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2013. No. 1307.5522.

This is an expanded version of my talk at the workshop ``Groups of Automorphisms in Birational and Affine Geometry'', October 29–November 3, 2012, Levico Terme, Italy. The first section is focused on Jordan groups in abstract setting, the second on that in the settings of automorphisms groups and groups of birational self-maps of algebraic varieties. ...

Added: July 21, 2013

Roman Avdeev, Petukhov A., Transformation Groups 2021 Vol. 26 No. 3 P. 719–774

Let G be a symplectic or special orthogonal group, let H be a connected reductive subgroup of G, and let X be a flag variety of G. We classify all triples (G, H, X) such that the natural action of H on X is spherical. For each of these triples, we determine the restrictions to ...

Added: September 2, 2020

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2021. No. 2105.12861.

Starting with exploration of the possibility to present the underlying variety of an affine algebraic group in the form of a product of some algebraic varieties, we then explore the naturally arising problem as to what extent the group variety of an algebraic group determines its group structure. ...

Added: May 28, 2021

Yuri Prokhorov, Constantin Shramov, Compositio Mathematica 2014 Vol. 150 No. 12 P. 2054–2072

Assuming a particular case of the Borisov–Alexeev–Borisov conjecture, we prove that finite subgroups of the automorphism group of a finitely generated field over Q have bounded orders. Further, we investigate which algebraic varieties have groups of birational selfmaps satisfying the Jordan property. Unless explicitly stated otherwise, all varieties are assumed to be algebraic, geometrically irreducible ...

Added: January 29, 2015

Shirokov D., Advances in Applied Clifford Algebras 2021 Vol. 31 Article 30

In this paper, we consider inner automorphisms that leave invariant fixed subspaces of real and complex Clifford algebras — subspaces of fixed grades and subspaces determined by the reversion and the grade involution. We present groups of elements that define such inner automorphisms and study their properties. Some of these Lie groups can be interpreted ...

Added: May 10, 2021

Roman Avdeev, Selecta Mathematica, New Series 2015 Vol. 21 No. 3 P. 931–993

A subgroup H of an algebraic group G is said to be strongly solvable if H is contained in a Borel subgroup of G. This paper is devoted to establishing relationships between the following three combinatorial classifications of strongly solvable spherical subgroups in reductive complex algebraic groups: Luna’s general classification of arbitrary spherical subgroups restricted ...

Added: July 8, 2015

Савельева А. О., / Cornell University. Series math "arxiv.org". 2021.

We prove that the automorphism groups of Hopf manifolds are Jordan. ...

Added: November 24, 2021

Vladimir L. Popov, Springer Proceedings in Mathematics & Statistics 2014 Vol. 79 P. 185–213

This is an expanded version of my talk at the workshop
``Groups of Automorphisms in Birational and Affine Geometry'',
October 29–November 3, 2012, Levico Terme, Italy.
The first section is focused on Jordan groups in abstract setting,
the second on that in the settings of automorphisms groups and
groups of birational self-maps of algebraic varieties.
The appendix is an expanded version ...

Added: April 28, 2014

Sheina K., Известия высших учебных заведений. Поволжский регион. Физико-математические науки 2021 Т. 1 № 1 С. 49–65

The basic automorphism group of a Cartan foliation (M,F) is the quotient group of the automorphism group of (M, F ) by the normal subgroup, which preserves every leaf invariant. For Cartan foliations covered by fibrations, we find sufficient conditions for the existence of a structure of a finite-dimensional Lie group in their basic automorphism ...

Added: December 16, 2020

Vladimir L. Popov, Pacific Journal of Mathematics 2015 Vol. 279 No. 1--2 (Special issue In memoriam: Robert Steinberg) P. 423–446

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. ...

Added: December 27, 2015

V. L. Popov, Mathematical notes 2018 Vol. 103 No. 5 P. 811–819

We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This
implies that all algebraic (not necessarily affine) groups over fields of characteristic zero and some
transformation groups of complex spaces and Riemannian manifolds are Jordan. ...

Added: April 13, 2018

V. L. Popov, Doklady Mathematics 2017 Vol. 96 No. 1 P. 312–314

For connected simple algebraic groups defined over an algebraically closed field of characteristic
zero, the classifications of irreducible algebraic representations of modalities 0, 1, and 2 are obtained. ...

Added: June 30, 2017

Kham T., Социальные и гуманитарные науки: теория и практика 2019 № 1(3) С. 167–183

The article examines the problems of defining the term computer simulations of scientific experiments. The first part analyzes the original method for classifying variations of terms proposed by Duran as the most successful for demonstrating significant existing contradictions among philosophers regarding the place and role of computer simulations in the philosophy of science. In the ...

Added: December 11, 2019

Sheina K., / Cornell University. Series arXiv "math". 2020. No. 04348v1.

The basic automorphism group of a Cartan foliation (M, F) is the quotient group of the automorphism group of (M, F) by the normal subgroup, which preserves every leaf invariant. For Cartan foliations covered by fibrations, we find sufficient conditions for the existence of a structure of a finite-dimensional Lie group in their basic automorphism groups. Estimates ...

Added: December 9, 2020

N. I. Zhukova, Journal of Geometry and Physics 2018 Vol. 132 P. 146–154

We present a new method of investigation of G-structures on orbifolds.
This method is founded on the consideration of a G-structure on an
n-dimensional orbifold as the corresponding transversal
structure of an associated foliation. Using this method we prove the
existence and the uniqueness of a finite dimensional Lie group structures
on the full automorphism group of an elliptic G-structure ...

Added: April 4, 2017

Группы базовых автоморфизмов картановых слоений моделируемых на неэффективных картановых геометриях.

Zhukova N., Sheina K., Труды Математического центра им. Н.И. Лобачевского 2015 Т. 52 С. 73–74

Исследуются картановы слоения, то есть слоения допускающие трансверсальную картанову геометрию. Рассматривается общая ситуация, когда картанова геометрия может быть неэффективной. Найдено достаточное условие для того, чтобы полная группа базовых автоморфизмов картанова слоения со связностью Эресмана допускала единственную структуру конечномерной группы Ли в категории картановых слоений, где изоморфизмы сохраняют как слоение, так и трансверсальную геометрию. Получены некоторые ...

Added: October 14, 2015

/ Cornell University. Series arXiv "math". 2015. No. 1410.1144.

We get sufficient conditions for the full basic automorphism group of a complete Cartan foliation to admit a unique (finite-dimensional) Lie group structure in the category of Cartan foliations. In particular, we obtain sufficient conditions for this group to be discrete. Emphasize that the transverse Cartan geometry may be noneffective. Some estimates of the dimension ...

Added: September 28, 2015