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## Basic automorphism of Cartan foliations covered by fibrations

Cornell University
,
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 of the dimension of these groups are obtained. For some class of Cartan foliations with integrable an Ehresmann connection, a method for finding of basic automorphism groups is specified.

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

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

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

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

/ 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

Dolgonosova A., Журнал Средневолжского математического общества 2017 Т. 19 № 1 С. 19-29

The subject of this article is a review of the results on foliations with transversal linear connection obtained by the author together with N.I. Zhukova, and their comparison with the results of other authors. The work consists of three parts. The first part focuses on to automorphism groups of foliations with a transversal linear connection ...

Added: June 13, 2017

Gorinov A., Nikolay Konovalov, / Cornell University. Series "Working papers by Cornell University". 2020. No. 1712.02578.

Let $G$ be a complex connected reductive algebraic group that acts on a smooth complex algebraic variety $X$, and let $E$ be a $G$-homogeneous algebraic vector bundle over $X$. A section of $E$ is {\it regular} if it is transversal to the zero section. Let $U\subset\Gamma(X,E)$ be the subset of regular sections. We give a ...

Added: March 16, 2020

Dolgonosova A., Zhukova N., Журнал Средневолжского математического общества 2015 Т. 17 № 4 С. 14-23

We prove the equivalence of three different approaches to the definition of completeness of a foliation with transverse linear connection. It is shown that for the transverse ane foliations
(M, F) of codimension q, q > 1, each of the mentioned above conditions are equivalent to
fulllment of the following two conditions: 1) there exists an Ehresmann ...

Added: March 12, 2016

Aleksei Golota, International Journal of Mathematics 2020 Vol. 31 No. 10 P. 2050077

For a variety 𝑋, a big ℚ-divisor 𝐿 and a closed connected subgroup 𝐺⊂Aut(𝑋,𝐿) we define a 𝐺-invariant version of the 𝛿-threshold. We prove that for a Fano variety (𝑋,−𝐾_𝑋) and a connected subgroup 𝐺⊂Aut(𝑋) this invariant characterizes 𝐺-equivariant uniform 𝐾-stability. We also use this invariant to investigate 𝐺-equivariant 𝐾-stability of some Fano varieties with ...

Added: September 25, 2020

Sheina K., Zhukova N., Lobachevskii Journal of Mathematics 2016

For a complete Cartan foliation $(M,F)$ we introduce two algebraic invariants $\frak{g}_{0}(M,F)$ and
${\frak g}_{1}(M,F)$ which we
call structure Lie
algebras. If the transverse Cartan geometry of $(M,F)$ is effective then
$\frak{g}_{0}(M,F)={\frak g}_{1}(M,F)$. We prove that if $\frak{g}_{0}(M,F)$
is zero then in the category of Cartan foliations the group of all basic automorphisms of the ...

Added: October 12, 2016

Н.И. Жукова, Шеина К. И., Труды Математического центра им. Н.И. Лобачевского 2014 Т. 50 С. 74-76

We investigate Cartan foliations covered by fibrations. We obtain a sufficient condition for the full
basic automorphism group of a complete Cartan foliation covered by fibration to admit a
unique (finite-dimensional) Lie group structure in the category of
Cartan foliations. The explicit new formula for determining its basic automorphism
Lie group is given. Examples of computing the full basic ...

Added: November 12, 2014

Nikolay Konovalov, / Cornell University. Series "Working papers by Cornell University". 2022. No. 2202.07507.

Let $V_{n,d}$ be the variety of equations for hypersurfaces of degree $d$ in $\mathbb{P}^n(\mathbb{C})$ with singularities not worse than simple nodes. We prove that the orbit map $G'=SL_{n+1}(\mathbb{C}) \to V_{n,d}$, $g\mapsto g\cdot s_0$, $s_0\in V_{n,d}$ is surjective on the rational cohomology if $n>1$, $d\geq 3$, and $(n,d)\neq (2,3)$. As a result, the Leray-Serre spectral sequence ...

Added: September 12, 2022

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., Chebochko N., Известия высших учебных заведений. Математика 2020 № 11 С. 87-92

The aim of this work is to describe the structure of complete Lorentzian foliations $(M, F)$ of codimension two
on $n$-dimensional closed manifolds. It is proved that $(M, F)$ is either Riemannian or has a constant
transversal curvature and its structure is described. For such foliations $(M, F)$, the criterion is obtained,
reducing the chaos problem in $(M, ...

Added: October 6, 2020

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

Zhukova N., Журнал Средневолжского математического общества 2018 Т. 20 № 4 С. 395-407

It is shown that the structural theory of Molino for Riemannian foliations on compact
manifolds and complete Riemannian manifolds is generalized to Riemannian foliations with
Ehresmann connection. There are no restrictions on the codimension of the foliation
and the dimension of the foliated manifold.
For a Riemannian foliation $(M, F)$ with Ehresmann connection
it is proved that the closure of ...

Added: December 27, 2019

Багаев А. В., Zhukova N., Труды математического центра имени Н.И. Лобачевского 2017

We prove a criterion for Lorentzian foliations of codimension two with
Ehresmann connection to be Riemannian. A description of the structure of transverse analytic
non-Riemannian Lorentzian foliations of codimension two is given. ...

Added: November 16, 2017

Багаев А. В., Zhukova N., Известия высших учебных заведений. Поволжский регион. Физико-математические науки 2017 № 4 (44) С. 35-47

Actuality and goals. The Lorentzian geometry is radically different from the Riemannian geometry and finds widespread application in various physical theories. The goal of this work is to investigate the structure of transversely analytical Lorentzian foliations (M,F) of codimension two on n-dimensional manifolds.
Methods. The methods of foliated bundles and holonomy pseudogroups are applied.
Results. We prove ...

Added: November 15, 2017

Vladimir L. Popov, Transformation Groups 2014 Vol. 19 No. 2 P. 549-568

We explore orbits, rational invariant functions, and quotients of the natural actions of connected, not necessarily finite dimensional subgroups of the automorphism groups of irreducible algebraic varieties. The applications of the results obtained are given. ...

Added: March 17, 2014

Popov V. L., Zarhin Y., / Cornell University. Series math "arxiv.org". 2018. No. 1808.01136.

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 $\mathcal L(K)$ generated by ${\rm Aut} (K)$ and multipli\-ca\-tions by the elements of $K^*$. We also classify the Weyl groups of roots systems of rank $n$ which are ...

Added: August 8, 2018

Prokhorov Y., Shramov K., / Cornell University. Series arXiv "math". 2018.

We prove that automorphism groups of Inoue and primary Kodaira surfaces are Jordan. ...

Added: June 8, 2019

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

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