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73
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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Cited by 79 (17 self)
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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,
Some properties of finitedimensional semisimple Hopf algebras
, 1998
"... Kaplansky conjectured that if H is a finitedimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is of Frobenius type (i.e. if V is an irreducible representation of H then dimV divides dimH) [Ka]. It was proved that the conjecture is true for H of dime ..."
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Cited by 37 (15 self)
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Kaplansky conjectured that if H is a finitedimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is of Frobenius type (i.e. if V is an irreducible representation of H then dimV divides dimH) [Ka]. It was proved that the conjecture is true for H of dimension p n, p prime [MW], and that if H has a 2−dimensional representation then dimH is even [NR]. In this paper we first prove in Theorem 1.4 that if V is an irreducible representation of D(H), the Drinfeld double of any finitedimensional semisimple Hopf algebra H over k, then dimV divides dimH (not just dimD(H) = (dimH) 2). In doing this we use the theory of modular tensor categories (in particular Verlinde formula). We then use it to prove in Theorem 1.5 that Kaplansky’s conjecture is true for finitedimensional semisimple quasitriangular Hopf algebras over k. As a result we prove easily in Theorem 1.7 that Kaplansky’s conjecture [Ka] on prime dimensional Hopf algebras over k is true by passing to their Drinfeld doubles (compare with [Z]). Second, we use a theorem of Deligne [De] to prove in Theorem 2.2 that triangular semisimple Hopf algebras over k are equivalent to group algebras as quasiHopf algebras [Dr2].
Triangular Hopf algebras with the Chevalley property
 Michigan Journal of Mathematics
"... Triangular Hopf algebras were introduced by Drinfeld [Dr]. They are the Hopf algebras whose representations form a symmetric tensor category. In that sense, they are the class of Hopf algebras closest to group algebras. The structure of triangular Hopf algebras is far from ..."
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Cited by 31 (11 self)
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Triangular Hopf algebras were introduced by Drinfeld [Dr]. They are the Hopf algebras whose representations form a symmetric tensor category. In that sense, they are the class of Hopf algebras closest to group algebras. The structure of triangular Hopf algebras is far from
From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
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Invariants of 3–manifolds and projective representations of mapping class groups via quantum groups at roots of unity
 Comm. Math. Phys
, 1995
"... Abstract. An example of a finite dimensional factorizable ribbon Hopf Calgebra is given by a quotient H = uq(g) of the quantized universal enveloping algebra Uq(g) at a root of unity q of odd degree. The mapping class group Mg,1 of a surface of genus g with one hole projectively acts by automorphis ..."
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Cited by 23 (1 self)
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Abstract. An example of a finite dimensional factorizable ribbon Hopf Calgebra is given by a quotient H = uq(g) of the quantized universal enveloping algebra Uq(g) at a root of unity q of odd degree. The mapping class group Mg,1 of a surface of genus g with one hole projectively acts by automorphisms in the Hmodule H ∗⊗g, if H ∗ is endowed with the coadjoint Hmodule structure. There exists a projective representation of the mapping class group Mg,n of a surface of genus g with n holes labelled by finite dimensional Hmodules X1,..., Xn in the vector space HomH(X1 ⊗ · · · ⊗ Xn, H ∗⊗g). An invariant of closed oriented 3manifolds is constructed. Modifications of these constructions for a class of ribbon Hopf algebras satisfying weaker conditions than factorizability (including most of uq(g) at roots of unity q of even degree) are described. After works of Moore and Seiberg [44], Witten [62], Reshetikhin and Turaev [51], Walker [61], Kohno [22, 23] and Turaev [59] it became clear that any semisimple abelian ribbon category with finite number of simple objects satisfying some nondegeneracy condition gives rise to projective representations of mapping class groups
Classification of finitedimensional triangular Hopf algebras with the Chevalley property
 Mathematical Research Letters
"... Abstract. A fundamental problem in the theory of Hopf algebras is the classification and construction of finitedimensional quasitriangular Hopf algebras over C. Quasitriangular Hopf algebras constitute a very important class of Hopf algebras, introduced by Drinfeld. They are the Hopf algebras whose ..."
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Cited by 21 (9 self)
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Abstract. A fundamental problem in the theory of Hopf algebras is the classification and construction of finitedimensional quasitriangular Hopf algebras over C. Quasitriangular Hopf algebras constitute a very important class of Hopf algebras, introduced by Drinfeld. They are the Hopf algebras whose representations form a braided tensor category. However, this intriguing problem is extremely hard and is still widely open. Triangular Hopf algebras are the quasitriangular Hopf algebras whose representations form a symmetric tensor category. In that sense they are the closest to group algebras. The structure of triangular Hopf algebras is far from trivial, and yet is more tractable than that of general Hopf algebras, due to their proximity to groups. This makes triangular Hopf algebras an excellent testing ground for general Hopf algebraic ideas, methods and conjectures. A general classification of triangular Hopf algebras is not known yet. However, the problem was solved in the semisimple case, in the minimal triangular pointed case, and more generally for triangular Hopf algebras with the Chevalley property. In this paper we report on all of this, and explain in full details the mathematics and ideas involved in this theory. The classification in the semisimple case relies on Deligne’s theorem on Tannakian categories and on Movshev’s theory in an essential way. We explain Movshev’s theory in details, and refer to [G5] for a detailed discussion of the first aspect. We also discuss the existence of grouplike elements in quasitriangular semisimple Hopf algebras, and the representation theory of cotriangular semisimple Hopf algebras. We conclude the paper with a list of open problems; in particular with the question whether any finitedimensional triangular Hopf algebra over C has the Chevalley property. 1.
On finitedimensional semisimple and cosemisimple Hopf algebras in positive characteristic
 Internat. Math. Res. Notices
, 1998
"... Recently, important progress has been made in the study of finitedimensional semisimple Hopf algebras over a field of characteristic zero (see [Mo] and references therein). Yet, very little is known over a field k of positive characteristic. In this paper we first prove in Theorem 2.1 that any fini ..."
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Cited by 20 (6 self)
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Recently, important progress has been made in the study of finitedimensional semisimple Hopf algebras over a field of characteristic zero (see [Mo] and references therein). Yet, very little is known over a field k of positive characteristic. In this paper we first prove in Theorem 2.1 that any finitedimensional semisimple and
On higher FrobeniusSchur indicators
"... We study the higher FrobeniusSchur indicators of modules over semisimple Hopf algebras, and relate them to other invariants as the exponent, the order, and the index. We prove various divisibility and integrality results for these invariants. Furthermore, we give some examples that illustrate the g ..."
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Cited by 17 (0 self)
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We study the higher FrobeniusSchur indicators of modules over semisimple Hopf algebras, and relate them to other invariants as the exponent, the order, and the index. We prove various divisibility and integrality results for these invariants. Furthermore, we give some examples that illustrate the general theory.
Factorizable ribbon quantum groups in logarithmic conformal field theories
 THEOR. MATH. PHYS
, 207
"... We review the properties of quantum groups occurring as Kazhdan–Lusztig dual to logarithmic conformal field theory models. These quantum groups at even roots of unity are not quasitriangular but are factorizable and have a ribbon structure; the modular group representation on their center coincides ..."
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Cited by 17 (9 self)
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We review the properties of quantum groups occurring as Kazhdan–Lusztig dual to logarithmic conformal field theory models. These quantum groups at even roots of unity are not quasitriangular but are factorizable and have a ribbon structure; the modular group representation on their center coincides with the representation on generalized characters of the chiral algebra in logarithmic conformal field models.