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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,
Weakly grouptheoretical and solvable fusion categories
"... To Izrail Moiseevich Gelfand on his 95th birthday with admiration ..."
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Cited by 17 (4 self)
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To Izrail Moiseevich Gelfand on his 95th birthday with admiration
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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Cited by 13 (3 self)
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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.
2007 Orders and dimensions for sl(2) or sl(3) module categories and Boundary Conformal Field Theories on a torus J Math Phys 48 p 043511 (preprint mathph/0610073
"... After giving a short description, in terms of action of categories, of some of the structures associated with sl(2) and sl(3) boundary conformal field theories on a torus, we provide tables of dimensions describing the semisimple and cosemisimple blocks of the corresponding weak bialgebras (quantum ..."
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Cited by 7 (4 self)
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After giving a short description, in terms of action of categories, of some of the structures associated with sl(2) and sl(3) boundary conformal field theories on a torus, we provide tables of dimensions describing the semisimple and cosemisimple blocks of the corresponding weak bialgebras (quantum groupoids), tables of quantum dimensions and orders, and tables describing induction restriction. For reasons of size, the sl(3) tables of induction are only given for theories with selffusion (existence of a monoidal structure).
From boundary to bulk in logarithmic CFT
, 2007
"... The analogue of the chargeconjugation modular invariant for rational logarithmic conformal field theories is constructed. This is done by reconstructing the bulk spectrum from a simple boundary condition (the analogue of the Cardy ‘identity brane’). We apply the general method to the c1,p triplet m ..."
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Cited by 6 (1 self)
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The analogue of the chargeconjugation modular invariant for rational logarithmic conformal field theories is constructed. This is done by reconstructing the bulk spectrum from a simple boundary condition (the analogue of the Cardy ‘identity brane’). We apply the general method to the c1,p triplet models and reproduce the previously known bulk theory for p = 2 at c = −2. For general p we verify that the resulting partition functions are modular invariant. We also construct the complete set of 2p boundary states, and confirm that the identity brane from which we started indeed exists. As a byproduct we obtain a logarithmic version of the Verlinde formula for the c1,p triplet models.
A FINITENESS PROPERTY FOR BRAIDED FUSION CATEGORIES
"... Abstract. We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has property F if the associated braid group representations ..."
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Cited by 5 (3 self)
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Abstract. We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has property F if the associated braid group representations factor over a finite group, and suggest that categories of integral FrobeniusPerron dimension are precisely those with property F. 1.
AN ANALOGUE OF RADFORD’S S 4FORMULA FOR FINITE TENSOR CATEGORIES
, 2004
"... Abstract. We develop the theory of Hopf bimodules for a finite rigid tensor category C. Then we use this theory to define a distinguished invertible object D of C and an isomorphism of tensor functors δ: V ∗ ∗ → D ⊗ ∗ ∗ V ⊗D −1. This provides a categorical generalization of Radford’s S 4 formula fo ..."
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Cited by 3 (1 self)
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Abstract. We develop the theory of Hopf bimodules for a finite rigid tensor category C. Then we use this theory to define a distinguished invertible object D of C and an isomorphism of tensor functors δ: V ∗ ∗ → D ⊗ ∗ ∗ V ⊗D −1. This provides a categorical generalization of Radford’s S 4 formula for finite dimensional Hopf algebras [R1], which was proved in [N] for weak Hopf algebras, in [HN] for quasiHopf algebras, and conjectured in general in [EO]. When C is braided, we establish a connection between δ and the Drinfeld isomorphism of C, extending the result of [R2]. We also show that a factorizable braided tensor category is unimodular (i.e. D = 1). Finally, we apply our theory to prove that the pivotalization of a fusion category is spherical, and give a purely algebraic characterization of exact module categories defined in [EO]. 1.
Formal proof, computation, and the construction problem in algebraic geometry
"... It has become a classical technique to turn to theoretical computer science to provide computational tools for algebraic geometry. A more recent transformation is that now we also get logical tools, and these too should be useful in the study of algebraic varieties. The purpose of this note is to co ..."
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It has become a classical technique to turn to theoretical computer science to provide computational tools for algebraic geometry. A more recent transformation is that now we also get logical tools, and these too should be useful in the study of algebraic varieties. The purpose of this note is to consider a very small part of this picture, and try to motivate the study of computer theoremproving techniques by looking at how they might be relevant to a particular class of problems in algebraic geometry. This is only an informal discussion, based more on questions and possible research directions than on actual results. This note amplifies the themes discussed in my talk at the “Arithmetic and Differential Galois Groups ” conference (March 2004, Luminy), although many specific points in the discussion were only finished more recently. I would like to thank: André Hirschowitz and Marco Maggesi, for their invaluable insights about computerformalized mathematics as it relates
On formal codegrees of fusion categories
 Math. Research Letters
, 2009
"... Abstract. We prove a general result which implies that the global and FrobeniusPerron dimensions of a fusion category generate Galois invariant ideals in the ring of algebraic integers. 1. ..."
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Abstract. We prove a general result which implies that the global and FrobeniusPerron dimensions of a fusion category generate Galois invariant ideals in the ring of algebraic integers. 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].