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On algebraic group varieties
Cornell University
,
2021.
No. 2102.08032.
Several results on presenting an affine algebraic group variety as a product of algebraic varieties are obtained.
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
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
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
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
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
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
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
Roman Avdeev, / Cornell University. Series math "arxiv.org". 2019. No. 1905.01169.
We obtain several structure results for a class of spherical subgroups of connected reductive complex algebraic groups that extends the class of strongly solvable spherical subgroups. Based on these results, we construct certain one-parameter degenerations of the Lie algebras corresponding to such subgroups. As an application, we exhibit an explicit algorithm for computing the set ...
Added: June 1, 2019
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
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
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, 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, 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
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
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
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
В. Л. Попов, Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2021 Т. 500 № 1 С. 52-54
It is explored to which extent the group variety of an algebraic group determines its group structure. ...
Added: November 18, 2021
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
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
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
Roman Avdeev, Petukhov A., Algebras and Representation Theory 2020 Vol. 23 No. 3 P. 541-581
Let G be a connected semisimple algebraic group and let H⊂G be a connected reductive subgroup. Given a flag variety X of G, a result of Vinberg and Kimelfeld asserts that H acts spherically on X if and only if for every irreducible representation R of G realized in the space of sections of a ...
Added: February 11, 2019
Popov V., Труды Математического института им. В.А. Стеклова РАН 2023 Т. 320 С. 287-297
Построена новая бесконечная серия рациональных аффинных алгебраических многообразий, группа автоморфизмов которых содержит группу Aut}(F_n) автоморфизмов свободной группы ранга n. Группы автоморфизмов таких многообразий нелинейны и содержат группу кос B_n c n нитями при n > 2, а при n > 1 неаменабельны. В качестве приложения доказано, что при n > 2 каждая группа Кремоны ранга ...
Added: June 9, 2022
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