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## On Quantizable Odd Lie Bialgebras

Motivated by the obstruction to the deformation quantization of Poisson structures in *infinite*dimensions, we introduce the notion of a quantizable odd Lie bialgebra. The main result of the paper is a construction of the highly non-trivial minimal resolution of the properad governing such Lie bialgebras, and its link with the theory of so-called *quantizable* Poisson structures.

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].

Our book is a selection of works presented at the Conference of Mathematical Physics “Kezenoi-Am 2016”. The Organising and Programme Committee of the conference tried to create a programme which embraces the variety of research directions inspired by the modern developments in Mathematical Physics and the theory of Integrable Systems. The authors of the included papers are well known mathematicians from several research groups in Europe and Russia. We hope that the book will attract the attention to these areas of research, and will be interesting both to experts and young researchers. The Conference of Mathematical Physics “Kezenoi-Am 2016” is the first in the series of conferences which are being held partly in the city of Grozny, and partly at a beautiful mountain lake “Kezenoi-Am” of the Chechen Republic, Russia. These conferences are generously supported by the Chechen State University (CheSU), which is driven by the goal to support mathematical culture in Chechen republic. It is important to note that the Chairman of the Organizing Committee, Rector of the CheSU, Prof. Zaurbek Saidov, encourages the idea that the organization of international conferences with the participation of world recognized researches is the optimal way to motivate and attract students to research. During the participants’ welcoming, he stressed out that he considers our conference to be a quite important step towards this direction. We are especially grateful to him for his support and help in the organization of this series of successful international conferences. We are also grateful to the Vice-rector of CheSU, Prof. Zaur Kindarov for his support and hospitality. Dr. Dmitry Grinev played an essential role in the organization of these conferences. We would like to thank him for his activity and patience throughout this procedure. Finally, we acknowledge that this work was carried out within the framework of the State Programme of the Ministry of Education and Science of the Russian Federation, project 1.12873.2018/12.1.

Victor M. Buchstaber, Sotiris Konstantinou-Rizos, Alexander V. Mikhailov

Yaroslavl, Russia June 2018

Based on a construction by Kashiwara and Rouquier, we present an analogue of the Beilinson-Bernstein localization theorem for hypertoric varieties. In this case, sheaves of differential operators are replaced by sheaves of W-algebras. As a special case, our result gives a localization theorem for rational Cherednik algebras associated to cyclic groups.

We apply the technique of formal geometry to give a necessary and sufficient condition for a line bundle supported on a smooth Lagrangian subvariety to deform to a sheaf of modules over a fixed deformation quantization of the structure sheaf of an algebraic symplectic variety.

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.

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.