Topic
Category O
About: Category O is a research topic. Over the lifetime, 306 publications have been published within this topic receiving 9643 citations.
Papers published on a yearly basis
Papers
TL;DR: In this paper, Soergel et al. showed that the block of the Bernstein-Gelfand-gelfand category O that corresponds to any fixed central character is a Koszul ring and the dual of that ring governs a certain subcategory of the category O again.
Abstract: The aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of Koszul duality to repre- sentation theory. The paper consists of three parts relatively independent of each other. The first part gives a reasonably selfcontained introduction to Koszul rings and Koszul duality. Koszul rings are certain Z-graded rings with particularly nice homological properties which involve a kind of duality. Thus, to a Koszul ring one associates naturally the dual Koszul ring. The second part is devoted to an application to representation theory of semisimple Lie algebras. We show License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use KOSZUL DUALITY PATTERNS 527 that the block of the Bernstein-Gelfand-Gelfand category O that corresponds to any fixed central character is governed by the Koszul ring. Moreover, the dual of that ring governs a certain subcategory of the category O again. This generalizes the selfduality theorem conjectured by Beilinson and Ginsburg in 1986 and proved by Soergel in 1990. In the third part we study certain cate- gories of mixed perverse sheaves on a variety stratified by affine linear spaces. We provide a general criterion for such a category to be governed by a Koszul ring. In the flag variety case this reduces to the setup of part two. In the more general case of affine flag manifolds and affine Grassmannians the criterion should yield interesting results about representations of quantum groups and affine Lie algebras. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 E-mail address: sasha@math.mit.edu Department of Mathematics, The University of Chicago, Chicago, Illinois 60637 E-mail address: ginzburg@math.uchicago.edu Max-Planck-Institut fur Mathematik, Gottfried-Claren-Strase 26, D-53 Bonn 3, Germany Current address: Mathematisches Institut, Universitat Freiburg, Albertstrase 23b, D-79104 Freiburg, Germany E-mail address: soergel@sun1.mathematik.uni-freiburg.de License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
1,201 citations
Journal Article•
TL;DR: In this article, the authors extend this connection with finite dimensional algebra representation theory into a central theme, which they call the tilting theory of finite-dimensional algebras.
Abstract: This paper continues the program begun by us in [8]), [9] (see also [15], [18]) in which the authors have begun to exploit in the modular representation theory of semisimple algebraic groups some of the powerful techniques of the theory of derived categories. As noted in the above references, the Inspiration for this work comes both from geometry, in the form of the classic algebraic work of Bernstein-Beilinson-Deligne [1] on singular spaces and perverse sheaves, and from the tilting theory of finite dimensional algebras [2], [3], [13], [14]. The present paper broadens and extends this connection with finite dimensional algebra representation theory into a central theme.
884 citations
Book•
1 Apr 1995
TL;DR: Lie Algebras. as mentioned in this paper The Weyl Group and its Geometry and Conjugacy Theorems of Kac-Moody Algebraic Lie Algebra.
Abstract: Lie Algebras. Lie Algebras Admitting Triangular Decompositions. Lattices and Root Systems. Contragredient Lie Algebras. The Weyl Group and Its Geometry. Category O for Kac--Moody Algebras. Conjugacy Theorems. Appendix. Bibliography. Index.
465 citations
TL;DR: In this article, the authors studied the category O of representations of the rational Cherednik algebra A attached to a complex reflection group W. They proved that the category of O is equivalent to the module category over a finite dimensional algebra, a generalized "q-Schur algebra" associated to W. The standard A-modules go, under this equivalence, to certain modules arising in Kazhdan-Lusztig theory of ''cells''.
Abstract: We study the category O of representations of the rational Cherednik algebra A attached to a complex reflection group W. We construct an exact functor, called Knizhnik-Zamolodchikov functor, from O to the category of H-modules, where H is the (finite) Iwahori-Hecke algebra associated to W. We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between O/O_tor, the quotient of O by the subcategory of A-modules supported on the discriminant and the category of finite-dimensional H-modules. The standard A-modules go, under this equivalence, to certain modules arising in Kazhdan-Lusztig theory of ``cells'', provided W is a Weyl group and the Hecke algebra H has equal parameters. We prove that the category O is equivalent to the module category over a finite dimensional algebra, a generalized "q-Schur algebra" associated to W.
339 citations
Posted Content•
TL;DR: In this article, the authors define and study sl\_2-categorifications on abelian categories, and show that there is a self-derived equivalence categorifying the adjoint action of the simple reflection.
Abstract: We define and study sl\_2-categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection. We construct categorifications for blocks of symmetric groups and deduce that two blocks are splendidly Rickard equivalent whenever they have isomorphic defect groups and we show that this implies Brou\'e's abelian defect group conjecture for symmetric groups. We give similar results for general linear groups over finite fields. The constructions extend to cyclotomic Hecke algebras. We also construct categorifications for category O of gl\_n(C) and for rational representations of general linear groups over an algebraically closed field of characteristic p, where we deduce that two blocks corresponding to weights with the same stabilizer under the dot action of the affine Weyl group have equivalent derived (and homotopy) categories, as conjectured by Rickard.
323 citations
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| Year | Papers |
|---|---|
| 2022 | 4 |
| 2021 | 11 |
| 2020 | 6 |
| 2019 | 17 |
| 2018 | 10 |
| 2017 | 21 |