Попов Владимир Леонидович

Finite subgroups of diffeomorphism groups

2013 в печати · PREPRINT · en

We prove (1) the existence of a noncompact smooth four-dimensional manifold M whose diffeomorphism group Diff(M) contains an isomorphic copy of every finitely presented group; (2) a finiteness theorem on finite simple subgroups of Diff(N) for compact manifolds N.

Алексей Николаевич Паршин (к семидесятилетию со дня рождения)

2013 · ARTICLE · ru

Юбилейная статья в честь академика РАН А.Н.Паршина.

Tori in the Cremona groups

2012 · PREPRINT · en

Для некоторых типов подгрупп в полных, в аффинных и в специальных аффинных группах Кремоны получены классификации с точностью до сопряженности. Доказана теорема об алгебраичности нормализатора таких подгрупп. В качестве приложения получены новые результаты в проблеме линеаризуемости: результаты Бялыницкого-Бирули 1966-67 годов обобщены на несвязный случай. Доказаны для n-мерных торов в аффинной грунте Кремоны и в специальной аффинной группе Кремоны рангов n. Для групп Кремоны вводятся и обсуждаются понятия разложения Жордана и простых чисел кручения.

Rationality and the FML invariant

2012 · PREPRINT · en

Мы строим контрпримеры к гипотезе рациональности, относя-щейся к новому варианту инварианта Макар-Лиманова, введен-ному в работе A. Liendo, Ga-actions of fiber type on affine T-vari-eties, J. Algebra 324 (2010), 3653–3665.

Problems for the problem session

2012 · PREPRINT · en

Некоторые проблемы о структуре групп Кремоны, сформули-рованные (с комментариями) автором на международной кон-ференции Бирациональная и аффинная геометрия, Левико Тер-ме (Тренто), 29.10.12--03.11.12

Cross-sections, quotients, and representation rings of semisimple algebraic groups

2011 · ARTICLE · en

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 universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.

Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?

2011 · ARTICLE · en

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 question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Инвариантные рациональные функции на полупростых алгебрах Ли и гипотеза Гельфанда-–Кириллова

2011 · CHAPTER · ru

On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties

2011 · CHAPTER · en

A simple method of constructing a big stock of algebraic varieties with trivial Makar-Limanov invariant is described, the Derksen invariant of some varieties is computed, the generalizations of the Makar-Limanov and Derksen invariants are introduced and discussed, and some results on the Jordan property of automorphism groups of algebraic varieties are obtained.

Discrete complex reflection groups

2010 · CHAPTER · en