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

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.

Vladimir L. Popov, Underlying varieties and group structures / 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

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

Roman Avdeev, Cupit-Foutou S., Advances in Mathematics 2018 Vol. 328 P. 1299-1352

Given a connected reductive algebraic group G and a finitely generated monoid Γ of dominant weights of G, in 2005 Alexeev and Brion constructed a moduli scheme M_Γ for multiplicity-free affine G-varieties with weight monoid Γ. This scheme is equipped with an action of an `adjoint torus' T_ad and has a distinguished T_ad-fixed point X_0. ...

Added: February 25, 2018

Sheina K., Basic automorphism of Cartan foliations covered by fibrations / 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

Р.С. Авдеев, Труды Московского математического общества 2010 Т. 71 С. 235-269

A spherical homogeneous space G/H of a connected semisimple algebraic group G is called excellent if it is quasi-affine and its weight semigroup is generated by disjoint linear combinations of the fundamental weights of the group G. All the excellent affine spherical homogeneous spaces are classified up to isomorphism. ...

Added: February 25, 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

V. L. Popov, Transformation Groups 2011 Vol. 16 No. 3 P. 827-856

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a
closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove
that in arbitrary G such a cross-section exists if and only if the ...

Added: March 16, 2013

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

Р.С. Авдеев, Математические заметки 2013 Т. 94 № 1 С. 22-35

For an arbitrary connected solvable spherical subgroup H of a connected semisimple algebraic group G, we compute the group N_G(H), the normalizer of H in G. Thereby we complete a classification of all (not necessarily connected) solvable spherical subgroups in semisimple algebraic groups. ...

Added: February 25, 2014

Vladimir L. Popov, The Jordan property for Lie groups and automorphism groups of complex spaces / Cornell University. Series math "arxiv.org". 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. ...

Added: April 3, 2018

Р.С. Авдеев, Труды Московского математического общества 2011 Т. 72 № 1 С. 5-62

We develop a structure theory of connected solvable spherical subgroups in semisimple algebraic groups. Based on this theory, we obtain an explicit classification of all such subgroups up to conjugacy. ...

Added: February 25, 2014

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., 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

Arzhantsev I., Celik D., Hausen J., Journal of Algebra 2013 Vol. 387 P. 87-98

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of afinitely generated algebra of invariants. ...

Added: November 13, 2013

Roman Avdeev, Transformation Groups 2021 Vol. 26 No. 2 P. 403-431

Given a connected reductive complex algebraic group G and a spherical subgroup H⊂G, the extended weight monoid Λˆ+G(G/H) encodes the G-module structures on spaces of regular sections of all G-linearized line bundles on G/H. Assuming that G is semisimple and simply connected and H is specified by a regular embedding in a parabolic subgroup P⊂G, ...

Added: September 9, 2021

Р.С. Авдеев, Петухов А. В., Математический сборник 2014 Т. 205 № 9 С. 3-48

For every finite-dimensional vector space V and every V-flag variety X we list all connected reductive subgroups in GL(V) acting spherically on X. ...

Added: October 22, 2014

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, Algebraic groups whose orbit closures contain only finitely many orbits / Cornell University. Series math "arxiv.org". 2017. No. 1707.06914 [math.AG].

We classify all connected affine algebraic groups G such that there are only finitely many G-orbits in every algebraic G-variety containing a dense open G-orbit. We also prove that G enjoys this property if and only if every irreducible algebraic G-variety X is modality-regular, i.e., the modality of X (in the sense of V. Arnol’d) ...

Added: July 24, 2017

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

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

Popov V., Известия РАН. Серия математическая 2022 Т. 86 № 5 С. 73-96

We explore to what extent the group variety of a connected algebraic group or the group manifold of a real Lie group determines its group structure. ...

Added: June 9, 2022

Vladimir L. Popov, Bass' triangulability problem / Cornell University. Series math "arxiv.org". 2015. No. 1504.03867.

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' Triangulability Problem is the affirmative. To this end we ...

Added: April 16, 2015

В. Л. Попов, Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2017 Т. 475 № 1 С. 14-16

Даны классификации неприводимых представлений простых алгебраических групп модальностей 0, 1 и 2. ...

Added: May 3, 2017