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Of all publications in the section: 24
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Article
Arzhantsev I., Timashev D. Transformation Groups. 2001. Vol. 6. No. 2. P. 101-110.
Article
Loktev S., Rybnikov L. G., Oblomkov A. et al. Transformation Groups. 2008. Vol. 13. No. 3. P. 541-556.
Article
Kuwabara T. Transformation Groups. 2015. Vol. 20. No. 2. P. 437-461 .

We study algebras constructed by quantum Hamiltonian reduction associated with symplectic quotients of symplectic vector spaces, including deformed preprojective algebras, symplectic reection algebras (rational Cherednik algebras), and quantization of hypertoric varieties introduced by Musson and Van den Bergh in [MVdB]. We determine BRST cohomologies associated with these quantum Hamiltonian reductions. To compute these BRST cohomologies, we make use of the method of deformation quantization (DQ-algebras) and F-action studied by Kashiwara and Rouquier in [KR], and Gordon and Losev in [GL].

Article
Khoroshkin A. Transformation Groups. 2016. Vol. 21. No. 2. P. 479-518.

We prove the conjecture by Feigin, Fuchs and Gelfand describing the Lie algebra cohomology of formal vector fields on an  n-dimensional space with coefficients in symmetric powers of the coadjoint representation. We also compute the cohomology of the Lie algebra of formal vector fields that preserve a given flag at the origin. The latter encodes characteristic classes of flags of foliations and was used in the formulation of the local Riemann-Roch Theorem by Feigin and Tsygan. Feigin, Fuchs and Gelfand described the first symmetric power and to do this they had to make use of a fearsomely complicated computation in invariant theory.  By the application of degeneration theorems of appropriate Hochschild-Serre spectral  sequences we avoid the need to use the methods of FFG, and moreover we are able to describe all the symmetric powers at once.

Article
Khoroshkin A. Transformation Groups. 2015. P. 1-40.

We prove the conjecture by Feigin, Fuchs and Gelfand describing the Lie algebra cohomology of formal vector fields on an  n-dimensional space with coefficients in symmetric powers of the coadjoint representation. We also compute the cohomology of the Lie algebra of formal vector fields that preserve a given flag at the origin. The latter encodes characteristic classes of flags of foliations and was used in the formulation of the local Riemann-Roch Theorem by Feigin and Tsygan. Feigin, Fuchs and Gelfand described the first symmetric power and to do this they had to make use of a fearsomely complicated computation in invariant theory.  By the application of degeneration theorems of appropriate Hochschild-Serre spectral  sequences we avoid the need to use the methods of FFG, and moreover we are able to describe all the symmetric powers at once.

Article
Khoroshkin S. M., Nazarov M. Transformation Groups. 2018. Vol. 23. No. 1. P. 119-147.

We study canonical intertwining operators between induced modules of the trigonometric Cherednik algebra. We demonstrate that these operators correspond to the Zhelobenko operators for the affine Lie algebra . To establish the correspondence, we use the functor of Arakawa, Suzuki and Tsuchiya which maps certain modules to modules of the Cherednik algebra.

Article
Mukhin E., Feigin B. L., Loktev S. et al. Transformation Groups. 2001. Vol. 6. No. 1. P. 25-52.
Article
Feigin B. L., Kedem R., Loktev S. et al. Transformation Groups. 2001. Vol. 6. No. 1. P. 25-52.
Article
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 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.

Article
Kiritchenko Valentina, Krishna A. Transformation Groups. 2013. Vol. 18. No. 2. P. 391-413.

We obtain an explicit presentation for the equivariant cobordism ring of a complete flag variety. An immediate corollary is a Borel presentation for the ordinary cobordism ring. Another application is an equivariant Schubert calculus in cobordism. We also describe the rational equivariant cobordism rings of wonderful symmetric varieties of minimal rank.

Article
Feigin E., Fourier G., Littelmann P. Transformation Groups. 2017. Vol. 22. No. 2. P. 321-352.

We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree on the PBW basis of the corresponding Lie algebra, the other by fixing a homogeneous monomial order on the PBW basis.

In the favourable case a basis of the module is parametrized by the lattice points of a normal polytope. The filtrations induce at degenerations of the corresponding ag variety to its abelianized version and to a toric variety, the special fibres of the degenerations being projectively normal and arithmetically Cohen-Macaulay. The polytope itself can be recovered as a Newton-Okounkov body. We conclude the paper by giving classes of examples for favourable modules.

Article
Braverman A., Finkelberg M. V. Transformation Groups. 2005. Vol. 10. No. 3-4. P. 1-23.
Article
Kishimoto T., Yuri Prokhorov, Zaidenberg M. Transformation Groups. 2013. Vol. 18. No. 4. P. 1137-1153.

We give a criterion of existence of a unipotent group action on the affine cone over a projective variety or, more generally, on the affine quasicone over a variety which is projective over another affine variety.

Article
Glutsyuk A. Transformation Groups. 2011. Vol. 16. No. 2. P. 413-479.

We study finitely-generated nondiscrete free  subgroups  in Lie groups. We address the following question first raised by Etienne Ghys: is it always possible to make arbitrarily small perturbation of the generators of the free subgroup in such a way that the new group formed by the perturbed generators be not free? In other words,  is it possible to approximate generators of a free subgroup  by elements  satisfying a nontrivial relation? We prove that the answer to Ghys' question is  positive and generalize this result to certain non-free subgroups. We also consider the question on the best approximation rate in terms of the minimal length of relation in the approximating group. We give an upper bound on the optimal approximation rate that is exponential in the  r-th power of the minimal length of relation, where 0.19<r<0.2.

Article
Vladimir L. Popov. Transformation Groups. 2008. Vol. 13. No. 3--4. P. 819-837.
Article
Valentina Kiritchenko. Transformation Groups. 2017. Vol. 22. No. 2. P. 387-402.

We compute the Newton-Okounkov bodies of line bundles on the complete flag variety of GL_n for a geometric valuation coming from a flag of translated Schubert subvarieties. The Schubert subvarieties correspond to the terminal subwords in the decomposition (s_1)(s_2s_1)(s_3s_2s_1)(...)(s_{n-1}...s_1) of the longest element in the Weyl group. The resulting Newton-Okounkov bodies coincide with the Feigin-Fourier-Littelmann-Vinberg polytopes in type A.

Article
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.

Article
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. Finally, we provide examples of non-reduced M_Γ.