S. Sinel'shchikov, A. Stolin, L. Vaksman
We consider Knapp-Vogan Hecke algebras in the quantum group setting. This allows us to produce a quantum analogue of the Bernstein functor as a first step towards the cohomological induction for quantum groups.
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We give an analogue of the Hom functor and prove a generalized form of the nuclear democracy theorem of Tsuchiya and Kanie by using a notion of tensor product for two modules for a vertex operator algebra.
Theory of the quantized flag manifold as a quasi-scheme (non-commutative scheme) has been developed by Lunts-Rosenberg. They have formulated an analogue of the Beilinson-Bernstein correspondence using the $q$-differential operators introduced in their earlier paper. In this paper we shall establish its modified version using a class of $q$-differential operators, which is (possibly) smaller than the one used by Lunts-Rosenberg.
We use Kazhdan-Lusztig tensoring to, first, describe annihilating ideals of highest weight modules over an affine Lie algebra in terms of the corresponding VOA and, second, to classify tilting functors, an affine analogue of projective functors known in the case of a simple Lie algebra. For the sake of completeness, the classification of annihilating ideals is borrowed from our previous work, q-alg/9711011; the part on tilting functors is new.
We present a simple proof of a strengthening of the derived Beilinson-Bernstein localization theorem using the formalism of descent in derived algebraic geometry. The arguments and results apply to arbitrary modules without the need to fix infinitesimal character. Roughly speaking, we demonstrate that all Ug-modules are the invariants, or equivalently coinvariants, of the action of intertwining functors (a refined form of Weyl group symmetry). This is a quantum version of descent for the Grothendieck-Springer simultaneous resolution. In an appendix we present an alternative perspective, which …
The analogy between Yetter's deformation theory form (lax) monoidal functors and Gerstenahaber's deformation theory for associative algebras is solidified by shown that under reasonable conditions the category of functors with an action of a lax monoidal functor is abelian, that an analogue of the Hochschild cohomology of an algebra with coefficients in a bimodule exists for monoidal functors, and is given by right derived functors. The deformation cohmology of a monoidal natural transformation is shown to be a special case.
The functors constructed by Arakawa and the author relate the representation theory of gl_n and that of the degenerate affine Hecke algebra H_l of GL_l. They transform the Verma modules over gl_n to the standard modules over H_l. They transform the simple modules to the simple modules. We also prove that they transform the Jantzen filtration on the Verma modules to that on the standard modules. We obtain the following results for the representations of H_l by translating the corresponding results for gl_n through the functors: (i) the (generalized) Bernstein-Gelfand-Gelfand resolution for a ce…
We introduce a $p$-adic analytic analogue of Backelin and Kremnizer's construction of the quantum flag variety of a semisimple algebraic group, when $q$ is not a root of unity and $| q-1|<1$. We then define a category of $λ$-twisted $D$-modules on this analytic quantum flag variety. We show that when $λ$ is regular and dominant and when the characteristic of the residue field does not divide the order of the Weyl group, the global section functor gives an equivalence of categories between the coherent $λ$-twisted $D$-modules and the category of finitely generated modules over $\widehat{U_q^λ}$…
Derived deformation functors, Koszul duality, and Maurer-Cartan spaces
本文总结了Hinich的导出Maurer-Cartan函子如何建立微分分次李代数与Artinian微分分次交换代数上的导出Schlessinger函子间的等价关系，并探讨了形变问题及更一般Koszul对偶operad对的类似物。
Finite induction functor for vertex operator algebras
本文为顶点算子代数的嵌入引入有限诱导函子，证明其满足 Frobenius 互反律等性质，并明确计算了标准子代数和抛物型子代数上不可约模的有限诱导。
We introduce an analogue of the composition of the Cherednik and Drinfeld functor for twisted Yangians. Our definition is based on the Howe duality, and originates from the centralizer construction of twisted Yangians due to Olshanski. Using our functor, we establish a correspondence between intertwining operators on the tensor products of certain modules over twisted Yangians, and the extremal cocycle on the hyperoctahedral group.
We introduce a skew analogue of the composition of the Cherednik and Drinfeld functors for twisted Yangians. Our definition is based on the skew Howe duality, and originates from the centralizer construction of twisted Yangians due to Olshanski. Using our functor, we establish a correspondence between intertwining operators on the tensor products of certain modules over twisted Yangians, and the extremal cocycle on the hyperoctahedral group.
Let $H$ be a generic affine Hecke algebra (Iwahori-Matsumoto definition) over a polynomial algebra with a finite number of indeterminates over the ring of integers. We prove the existence of an integral Bernstein-Lusztig basis related to the Iwahori-Matsumoto basis by a strictly upper triangular matrix, from which we deduce that the center $Z$ of $H$ is finitely generated and that $H$ is a finite type $Z$-module (this was proved after inversion of the parameters by Bernstein-Lusztig), and we give some applications to the theory of $H$-modules where the parameters act by 0. These results are re…
This paper is devoted to study of differential calculi over quadratic algebras, which arise in the theory of quantum bounded symmetric domains. We prove that in the quantum case dimensions of the homogeneous components of the graded vector spaces of k-forms are the same as in the classical case. This result is well-known for quantum matrices. The quadratic algebras, which we consider in the present paper, are q-analogues of the polynomial algebras on prehomogeneous vector spaces of commutative parabolic type. This enables us to prove that the de Rham complex is isomorphic to the dual of a quan…
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