arXiv subjects

Milen Yakimov

Publications and source records attributed to Milen Yakimov.

62 records · Page 4Linked to original sources

arXiv2004

Categories of modules over an affine Kac-Moody algebra and the Kazhdan-Lusztig tensor product

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)_κof 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_κin the notation of [KL1, KL2, KL3]).

math.RT↗

arXiv2004

Affine Jacquet functors and Harish-Chandra categories

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.

math.RT↗

arXiv2003

The prolate spheroidal phenomena and bispectrality

Landau, Pollak, Slepian, and Tracy, Widom discovered that certain integral operators with so called Bessel and Airy kernels possess commuting differential operators and found important applications of this phenomena in time-band limiting and random matrix theory. In this paper we announce that very large classes of integral operators derived from bispectral algebras of rank 1 and 2 (parametrized by lagrangian grassmannians of infinitely large size) have this property. The above examples come from special points in these grassmannians.

math-ph↗

arXiv2001

Quantum invariant measures

We derive an explicit expression for the Haar integral on the quantized algebra of regular functions C_q[K] on the compact real form K of an arbitrary simply connected complex simple algebraic group G. This is done in terms of the irreducible *-representations of the Hopf *-algebra C_q[K]. Quantum analogs of the measures on the symplectic leaves of the standard Poisson structure on K which are (almost) invariant under the dressing action of the dual Poisson algebraic group K^* are also obtained. They are related to the notion of quantum traces for representations of Hopf algebras. As an application we define and compute explicitly quantum analogs of the Harish-Chandra c-functions associated to the elements of the Weyl group of G.

math.QA↗

arXiv2000

Discrete bispectral Darboux transformations from Jacobi operators

We construct families of bispectral difference operators of the form a(n)T + b(n) + c(n) T^{-1} where T is the shift operator. They are obtained as discrete Darboux transformations from appropriate extensions of Jacobi operators. We conjecture that along with operators previously constructed by Grunbaum, Haine, Horozov, and Iliev they exhaust all bispectral regular (i.e. a(n)c(n) \neq 0, for all integer n) operators of the form above.

math.CA↗

arXiv2000

Deformation Quantization of Lagrangian Fiber Bundles

Let (M, \om) be a symplectic manifold. A Lagrangian fiber bundle π: M -> B determines a completely integrable system on M. First integrals of this system are the pull-backs of functions on the base of the bundle. We show that for each Lagrangian fiber bundle πthere exist star products on C^\infty(M)[[h]] which do not deform the pointwise multiplication on the subalgebra π^*(C^\infty (B)) [[h]]. The set of equivalence classes of such star products is in bijection with formal deformations of the symplectic structure \om for which π: M -> B remains Lagrangian taken modulo formal symplectomorphisms of M.

math.QA↗

arXiv2000

The Double and Dual of a Quasitriangular Lie Bialgebra

Let G be a connected, simply connected Poisson-Lie group with quasitriangular Lie bialgebra g. An explicit description of the double D(g) is given, together with the embeddings of g and g^*. This description is then used to provide a construction of the double D(G). The aim of this work is to describe D(G) in sufficient detail to be able to apply the procedures of Semenov-Tian-Shansky and Drinfeld for the classification of symplectic leaves and Poisson homogeneous spaces for Poisson-Lie groups.

math.QA↗

arXiv2000

Symplectic Leaves of Complex Reductive Poisson-Lie Groups

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.

math.QA↗