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On fusion categories
 Annals of Mathematics
"... Abstract. In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series ofC. For fusion categories with commutative Gr ..."
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Abstract. In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series ofC. For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two central series are centralizers of each other in the sense of M. Müger. Dedicated to Leonid Vainerman on the occasion of his 60th birthday 1. introduction The theory of fusion categories arises in many areas of mathematics such as representation theory, quantum groups, operator algebras and topology. The representation categories of semisimple (quasi) Hopf algebras are important examples of fusion categories. Fusion categories have been studied extensively in the literature,
Perverse sheaves on affine flags and Langlands dual group
"... Abstract. This is the first in a series of papers devoted to describing the category of sheaves on the affine flag manifold of a (split) simple group in terms the Langlands dual group. In the present paper we provide such a description for categories which are geometric counterparts of a maximal com ..."
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Abstract. This is the first in a series of papers devoted to describing the category of sheaves on the affine flag manifold of a (split) simple group in terms the Langlands dual group. In the present paper we provide such a description for categories which are geometric counterparts of a maximal commutative subalgebra in the Iwahori Hecke algebra H; of the antispherical module for H; and of the space of Iwahoriinvariant Whittaker functions. As a byproduct we obtain some new properties of central sheaves introduced in [G]. Acknowledgements. This project was conceived during the IAS special year in Representation Theory (1998/99) led by G. Lusztig, as a result of conversations with D. Gaitsgory, M. Finkelberg and I. Mirkovic. The outcome was strongly influenced by conversations with A. Beilinson and V. Drinfeld. The stimulating interest of A. Braverman, D. Kazhdan, G. Lusztig and V. Ostrik was crucial for keeping the project alive. We are very grateful to all these people. We thank I. Mirkovic and D. Gaitsgory for the permission to use their unpublished results; and M. Finkelberg and D. Gaitsgory for taking the trouble to read the text and point out various lapses in the exposition. The second author was supported by NSF and Clay Institute. 1.
Quasiexceptional sets and equivariant coherent sheaves on the nilpotent cone, Represent. Theory 7
, 2003
"... sheaves on the nilcone of a simple complex algebraic group is introduced by the author in the paper Perverse coherent sheaves (the socalled perverse tstructure corresponding to the middle perversity). In the present note we show that the same tstructure can be obtained from a natural quasiexcep ..."
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Cited by 15 (4 self)
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sheaves on the nilcone of a simple complex algebraic group is introduced by the author in the paper Perverse coherent sheaves (the socalled perverse tstructure corresponding to the middle perversity). In the present note we show that the same tstructure can be obtained from a natural quasiexceptional set generating this derived category. As a consequence we obtain a bijection between the sets of dominant weights and pairs consisting of a nilpotent orbit, and an irreducible representation of the centralizer of this element, conjectured by Lusztig and Vogan (and obtained by other means by the author in the paper On tensor categories attached to cells in affine Weyl groups, tobe published). 1.
Grouptheoretical properties of nilpotent modular categories, eprint arXiv:0704.0195v2 [math.QA
"... Abstract. We characterize a natural class of modular categories of prime power FrobeniusPerron dimension as representation categories of twisted doubles of finite pgroups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects o ..."
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Abstract. We characterize a natural class of modular categories of prime power FrobeniusPerron dimension as representation categories of twisted doubles of finite pgroups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects ofC have integral FrobeniusPerron dimensions then C is grouptheoretical in the sense of [ENO]. As a consequence, we obtain that semisimple quasiHopf algebras of prime power dimension are grouptheoretical. Our arguments are based on a reconstruction of twisted group doubles from Lagrangian subcategories of modular categories (this is reminiscent to the characterization of doubles of quasiLie bialgebras in terms of Manin pairs given in [Dr]). 1. introduction In this paper we work over an algebraically closed field k of characteristic 0. By a fusion category we mean a klinear semisimple rigid tensor category C with finitely many isomorphism classes of simple objects, finite dimensional spaces of morphisms, and such that the unit object 1 of C is simple. We refer the reader to [ENO] for a general theory of such categories. A fusion category is pointed if all its simple objects are invertible. A pointed fusion category is equivalent to Vec ω G, i.e., the category of Ggraded vector spaces with the associativity constraint given by some cocycle ω ∈ Z 3 (G, k × ) (here G is a finite group). 1.1. Main results. Theorem 1.1. Any braided nilpotent fusion category has a unique decomposition into a tensor product of braided fusion categories whose FrobeniusPerron dimensions are powers of distinct primes. The notion of nilpotent fusion category was introduced in [GN]; we recall it in Subsection 2.2. Let us mention that the representation category Rep(G) of a finite group G is nilpotent if and only if G is nilpotent. It is also known that fusion categories of prime power FrobeniusPerron dimension are nilpotent [ENO]. On the other hand, Vec ω G is nilpotent for any G and ω. Therefore it is not true that any nilpotent fusion category is a tensor product of fusion categories of prime power dimensions.
Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group, preprint
"... Abstract. This paper is a continuation of [1]. In [1] we constructed an equivalence between the derived category of equivariant coherent sheaves on the cotangent bundle to the flag variety of a simple algebraic group and a (quotient of) the category of constructible sheaves on the affine flag variet ..."
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Cited by 8 (5 self)
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Abstract. This paper is a continuation of [1]. In [1] we constructed an equivalence between the derived category of equivariant coherent sheaves on the cotangent bundle to the flag variety of a simple algebraic group and a (quotient of) the category of constructible sheaves on the affine flag variety of the Langlands dual group. Below we prove certain properties of this equivalence; provide a similar “Langlands dual ” description for the category of equivariant coherent sheaves on the nilpotent cone; and deduce some conjectures by Lusztig and Ostrik. Acknowledgements. I am greatful to all the people mentioned in acknowledgements in [1]. I also thank Eric Sommers for stimulating interest. The author is supported by NSF and Clay Institute. 1. Statements 1.1. Recollection of notations and setup. We keep the setup and notations of [1]. In particular, Fℓ is the affine flag variety of a split simple group G over a field k which is either finite or algebraically closed; Wf is the Weyl group of G, and W is the extended affine Weyl group; fW f ⊂ fW ⊂ W are the sets of minimal length representatives of respectively 2sided and left cosets of Wf in W; PI is the category of Iwahori equivariant perverse sheaves on Fℓ is the category whose objects are mixed Iwahori equivariant perverse sheaves on Fℓ, and morphisms are weight 0 geometric morphisms, i.e. weight 0 morphisms between the pullbacks of sheaves to Fℓ ¯ k (the notation is (the notation used only for an algebraically closed k); while P mix I used for finite k only). Lw, w ∈ W are irreducible objects of PI, or irreducible selfdual objects of P mix on the cardinality of k. The Serre quotient categories f PI, f PI f P mix
Local systems on nilpotent orbits and weighted Dynkin diagrams, Represent. Theory 6
, 2002
"... Let G be a reductive algebraic group over the complex numbers, B a Borel subgroup of G, and T a maximal torus of B. We denote by Λ = Λ(G) the weight lattice of G with respect to T, and by Λ+ = Λ+(G) the set of dominant weights with respect to the positive roots defined by B. Let g be the Lie algebra ..."
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Cited by 4 (2 self)
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Let G be a reductive algebraic group over the complex numbers, B a Borel subgroup of G, and T a maximal torus of B. We denote by Λ = Λ(G) the weight lattice of G with respect to T, and by Λ+ = Λ+(G) the set of dominant weights with respect to the positive roots defined by B. Let g be the Lie algebra of G, and let N denote the nilpotent cone in g.
Cells in quantum affine sln
, 2003
"... ABSTRACT. We study Lusztig’s theory of cells for quantum affine sln. Using the geometric construction of the quantum group due to Lusztig and Ginzburg– Vasserot, we describe explicitly the twosided cells, the number of left cells in a two–sided cell, and the asymptotic algebra, verifying conjecture ..."
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Cited by 4 (2 self)
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ABSTRACT. We study Lusztig’s theory of cells for quantum affine sln. Using the geometric construction of the quantum group due to Lusztig and Ginzburg– Vasserot, we describe explicitly the twosided cells, the number of left cells in a two–sided cell, and the asymptotic algebra, verifying conjectures of Lusztig. 1.
An orderreversing duality map for conjugacy classes in Lusztig’s canonical quotient
 Transform. Groups
"... Abstract. We define a partial order on the set No,¯c of pairs (O, C), where O is a nilpotent orbit and C is a conjugacy class in Ā(O), Lusztig’s canonical quotient of A(O). We then construct an orderreversing duality map No,¯c → L No,¯c that satisfies many of the properties of the original Spaltens ..."
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Cited by 3 (2 self)
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Abstract. We define a partial order on the set No,¯c of pairs (O, C), where O is a nilpotent orbit and C is a conjugacy class in Ā(O), Lusztig’s canonical quotient of A(O). We then construct an orderreversing duality map No,¯c → L No,¯c that satisfies many of the properties of the original Spaltenstein duality map. This generalizes work of Sommers [16]. 1.
CALCULATING CANONICAL DISTINGUISHED INVOLUTIONS IN THE AFFINE WEYL GROUPS
, 2001
"... Abstract. Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential role in the KazhdanLusztig combinatorics of these groups. A distinguished involution is called canonical if it is the shortest element in its double coset with respect to the finite Weyl group. E ..."
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Abstract. Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential role in the KazhdanLusztig combinatorics of these groups. A distinguished involution is called canonical if it is the shortest element in its double coset with respect to the finite Weyl group. Each twosided cell in the affine Weyl group contains precisely one canonical distinguished involution. In this note we calculate the canonical distinguished involutions in the affine Weyl groups of rank ≤ 7. We also prove some partial results relating canonical distinguished involutions and Dynkin’s diagrams of the nilpotent orbits in the Langlands dual group. 1.
CELL 2REPRESENTATIONS OF FINITARY 2CATEGORIES
"... Abstract. We study 2representations of finitary 2categories with involution and adjunctions by functors on module categories over finite dimensional algebras. In particular, we define, construct and describe in detail (right) cell 2representations inspired by KazhdanLusztig cell modules for Heck ..."
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Abstract. We study 2representations of finitary 2categories with involution and adjunctions by functors on module categories over finite dimensional algebras. In particular, we define, construct and describe in detail (right) cell 2representations inspired by KazhdanLusztig cell modules for Hecke algebras. Under some natural assumptions we show that cell 2representations are strongly simple and do not depend on the choice of a right cell inside a twosided cell. This reproves and extends the uniqueness result on categorification of KazhdanLusztig cell modules for Hecke algebras of type A from [MS].