Abstract:
The development of the political processes in the countries of Central Asia, the Caspian Sea and Caucasian regions have always been dynamic and difficult to predict. These processes characterize the last 20 years with escalating dynamism, starting with the disintegration of the ex Soviet Union and origination of a number of sovereign countries. So in time an enormous in dimensions political geographical region was formed, including the lands from the Persian Gulf to the Ural and from the Caucasus to the Pamir Mountains, a region with a very large energy and geopolitical potential. In the conditions of constantly growing deficiency of energy resources, inevitably here were concentrated the attention and interests of the world-wide (USA, Russia, China) and regional (Turkey, Iran) players on the “Big chess-board”, as Zb. Brzezhinski successfully called them (1997).

Abstract:
To each simple Lie algebra g and an element w of the corresponding Weyl group De Concini, Kac and Procesi associated a subalgebra U^w_- of the quantized universal enveloping algebra U_q(g), which is a deformation of the universal enveloping algebra U(n_- \cap w(n_+)) and a quantization of the coordinate ring of the Schubert cell corresponding to w. The torus invariant prime ideals of these algebras were classified by M\'eriaux and Cauchon [25], and the author [30]. These ideals were also explicitly described in [30]. They index the the Goodearl-Letzter strata of the stratification of the spectra of U^w_- into tori. In this paper we derive a formula for the dimensions of these strata and the transcendence degree of the field of rational Casimirs on any open Richardson variety with respect to the standard Poisson structure [15].

Abstract:
We prove that the standard Poisson structure on the Grassmannian Gr(k, n) is invariant under the action of the Coxeter element c =(1 2 ... n). In particular, its symplectic foliation is invariant under c. As a corollary, we obtain a second, Poisson geometric proof of the result of Knutson, Lam, and Speyer that the Coxeter element c interchanges the Lusztig strata of Gr(k, n). We also relate the main result to known anti-invariance properties of the standard Poisson structures on cominuscule flag varieties.

Abstract:
All factorizable Lie bialgebra structures on complex reductive Lie algebras were described by Belavin and Drinfeld. We classify the symplectic leaves of the full class of corresponding connected Poisson-Lie groups. A formula for their dimensions is also proved.

Abstract:
To each category C of modules of finite length over a complex simple Lie algebra g, closed under tensoring with finite dimensional modules, we associate and study a category Aff(C)_\kappa of smooth modules (in the sense of Kazhdan and Lusztig [KL1]) of finite length over the corresponding affine Kac-Moody algebra in the case of central charge less than the critical level. Equivalent characterizations of these categories are obtained in the spirit of the works of Kazhdan-Lusztig [KL1] and Lian-Zuckerman [LZ1]. In the main part of this paper we establish a finiteness result for the Kazhdan-Lusztig tensor product which can be considered as an affine version of a theorem of Kostant [K]. It contains as special cases the finiteness results of Kazhdan, Lusztig [KL] and Finkelberg [F], and states that for any subalgebra f of g which is reductive in g the "affinization" of the category of finite length admissible (g, f) modules is stable under Kazhdan-Lusztig's tensoring with the "affinization" of the category of finite dimensional g modules (which is O_\kappa in the notation of [KL1, KL2, KL3]).

Abstract:
We define an affine Jacquet functor and use it to describe the structure of induced affine Harish-Chandra modules at noncritical levels, extending the theorem of Kac and Kazhdan [KK] on the structure of Verma modules in the Bernstein-Gelfand-Gelfand categories O for Kac-Moody algebras. This is combined with a vanishing result for certain extension groups to construct a block decomposition of the categories of affine Harish-Chandra modules of Lian and Zuckerman [LZ]. The latter provides an extension of the works of Rocha-Caridi, Wallach [RW] and Deodhar, Gabber, Kac [DGK] on block decompositions of BGG categories for Kac-Moody algebras. We also prove a compatibility relation between the affine Jacquet functor and the Kazhdan-Lusztig tensor product. A modification of this is used to prove that the affine Harish-Chandra category is stable under fusion tensoring with the Kazhdan-Lusztig category (a case of our finiteness result [Y]) and will be further applied in studying translation functors for Kac-Moody algebras, based on the fusion tensor product.

Abstract:
We classify the invariant prime ideals of a quantum partial flag variety under the action of the related maximal torus. As a result we construct a bijection between them and the torus orbits of symplectic leaves of the standard Poisson structure on the corresponding flag variety. It was previously shown by K. Goodearl and the author that the latter are precisely the Lusztig strata of the partial flag variety.

Abstract:
De Concini, Kac and Procesi defined a family of subalgebras U^w_+ of a quantized universal enveloping algebra U_q(g), associated to the elements of the corresponding Weyl group W. They are deformations of the universal enveloping algebras U(n_+ \cap Ad_w(n_-)) where n_\pm are the nilradicals of a pair of dual Borel subalgebras. Based on results of Gorelik and Joseph and an interpretation of U^w_+ as quantized algebras of functions on Schubert cells, we construct explicitly the H invariant prime ideals of each U^w_+ and show that the corresponding poset is isomorphic to W^{\leq w}, where H is the group of group-like elements of U_q(g). Moreover, for each H-prime of U^w_+ we construct a generating set in terms of Demazure modules related to fundamental representations.

Abstract:
In this note we prove the Launois-Lenagan conjecture on the classification of the automorphism groups of the algebras of quantum matrices R_q[M_n] of square shape for all positive integers n, base fields K, and deformation parameters q \in K^* which are not roots of unity.

Abstract:
We prove the Andruskiewitsch-Dumas conjecture that the automorphism group of the positive part of the quantized universal enveloping algebra $U_q({\mathfrak{g}})$ of an arbitrary finite dimensional simple Lie algebra g is isomorphic to the semidirect product of the automorphism group of the Dynkin diagram of g and a torus of rank equal to the rank of g. The key step in our proof is a rigidity theorem for quantum tori. It has a broad range of applications. It allows one to control the (full) automorphism groups of large classes of associative algebras, for instance quantum cluster algebras.