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## Превосходные аффинные сферические однородные пространства полупростых алгебраических групп

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.

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

Colliot-Thélène J., Kunyavskiĭ B., Vladimir L. Popov et al., Compositio Mathematica 2011 Vol. 147 No. 2 P. 428–466

Let k be a field of characteristic zero, let G be a connected reductive algebraic group
over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k rational functions on G, respectively, g. The conjugation action of G on itself induces
the adjoint action of G on g. We investigate the ...

Added: March 17, 2013

Р.С. Авдеев, Математический сборник 2012 Т. 203 № 11 С. 3–22

For an affine spherical homogeneous space G/H of a connected semisimple algebraic group G, we consider the factorization morphism by the action on G/H of a maximal unipotent subgroup of G. We prove that this morphism is equidimensional if and only if the weight semigroup of G/H satisfies a simple condition. ...

Added: February 25, 2014

Roman Avdeev, Cupit-Foutou S., Transformation Groups 2018 Vol. 23 No. 2 P. 299–327

We give a combinatorial description of all affine spherical varieties with prescribed weight monoid Γ. As an application, we obtain a characterization of the irreducible components of Alexeev and Brion’s moduli scheme M_Γ for such varieties. Moreover, we find several sufficient conditions for M_Γ to be irreducible and exhibit several examples where M_Γ is reducible. ...

Added: October 17, 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

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

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

Shabalin T., Сибирский математический журнал 2013 Т. 54 № 4 С. 947–958

Under study are the centralizers of 3-dimensional simple Lie subalgebras in the universal enveloping algebra of a 7-dimensional simple Malcev algebra. We find some sets of generators for these centralizers in characteristic not 2 nor 3 and for the subalgebra generated by the centralizer in the central closure of the universal enveloping algebra in characteristic ...

Added: September 16, 2014

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

Arzhantsev I., Liendo A., Stasyuk T., Journal of Pure and Applied Algebra 2021 Vol. 225 No. 2 P. 106499

Let X be a normal variety endowed with an algebraic torus action. An additive group action alpha on X is called vertical if a general orbit of alpha is contained in the closure of an orbit of the torus action and the image of the torus normalizes the image of alpha in Aut(X). Our first result in this paper ...

Added: July 29, 2020

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

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2021. No. 2102.08032.

Several results on presenting an affine algebraic group variety as a product of algebraic varieties are obtained. ...

Added: February 17, 2021

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

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

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

Switzerland: Birkhauser/Springer, 2019

Lie theory, inaugurated through the fundamental work of Sophus Lie during the late
nineteenth century, has proved central in many areas of mathematics and theoretical
physics. Sophus Lie’s formulation was originally in the language of analysis and
geometry; however, by now, a vast algebraic counterpart of the theory has been
developed. As in algebraic geometry, the deepest and most ...

Added: October 26, 2019

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

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

Р.С. Авдеев, Известия РАН. Серия математическая 2010 Т. 74 № 6 С. 3–26

The extended weight semigroup of a homogeneous space G/H of a connected semisimple algebraic group G characterizes the spectra of the representations of G on spaces of regular sections of homogeneous line bundles over G/H, including the space of regular functions on G/H. We compute the extended weight semigroups for all strictly irreducible affine spherical ...

Added: February 25, 2014

Roman Avdeev, Indagationes Mathematicae 2012 Vol. 23 No. 1-2 P. 10–18

In 1994, based on Roberts’ counterexample to Hilbert’s fourteenth problem, A’Campo-Neuen constructed an example of a linear action of a 12-dimensional commutative unipotent group H_0 on a 19-dimensional vector space V such that the algebra of invariants k[V]^{H_0} is not finitely generated. We consider a certain extension H of H_0 by a one-dimensional torus and ...

Added: February 25, 2014

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

Котенкова П.Ю., Математические заметки 2011 Т. 90 № 2 С. 269–279

В работе явно описаны классы GIT-эквивалентности линеаризованных линейных расслоений для диагональных действий линейных алгебраических групп SL(V)
и SO(V) на проективных многообразиях. ...

Added: September 17, 2015

Vladimir L. Popov, / 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