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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,
Lifting of quantum linear spaces and pointed Hopf algebras of order p3
 J Algebra
, 1998
"... Abstract. We propose the following principle to study pointed Hopf algebras, or more generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a Hopf algebra A, consider its coradical filtration and the associated graded coalgebra gr A. Then gr A is a graded Hopf algebra, since the c ..."
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Cited by 142 (19 self)
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Abstract. We propose the following principle to study pointed Hopf algebras, or more generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a Hopf algebra A, consider its coradical filtration and the associated graded coalgebra gr A. Then gr A is a graded Hopf algebra, since the coradical A0 of A is a Hopf subalgebra. In addition, there is a projection π: gr A → A0; let R be the algebra of coinvariants of π. Then, by a result of Radford and Majid, R is a braided Hopf algebra and gr A is the bosonization (or biproduct) of R and A0: gr A ≃ R#A0. The principle we propose to study A is first to study R, then to transfer the information to gr A via bosonization, and finally to lift to A. In this article, we apply this principle to the situation when R is the simplest braided Hopf algebra: a quantum linear space. As consequences of our technique, we obtain the classification of pointed Hopf algebras of order p 3 (p an odd prime) over an algebraically closed field of characteristic zero; with the same hypothesis, the characterization of the pointed Hopf algebras whose coradical is abelian and has index p or p 2; and an infinite family of pointed, nonisomorphic, Hopf algebras of the same dimension. This last result gives a negative
Finite tensor categories
 Moscow Math. Journal
"... These are lecture notes for the course 18.769 “Tensor categories”, taught by P. Etingof at MIT in the spring of 2009. In these notes we will assume that the reader is familiar with the basic theory of categories and functors; a detailed discussion of this theory can be found in the book [ML]. We wil ..."
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Cited by 75 (12 self)
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These are lecture notes for the course 18.769 “Tensor categories”, taught by P. Etingof at MIT in the spring of 2009. In these notes we will assume that the reader is familiar with the basic theory of categories and functors; a detailed discussion of this theory can be found in the book [ML]. We will also assume the basics of the theory of abelian categories (for a more detailed treatment see the book [F]). If C is a category, the notation X ∈ C will mean that X is an object of C, and the set of morphisms between X, Y ∈ C will be denoted by Hom(X, Y). Throughout the notes, for simplicity we will assume that the ground field k is algebraically closed unless otherwise specified, even though in many cases this assumption will not be needed. 1. Monoidal categories 1.1. The definition of a monoidal category. A good way of thinking
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 51 (16 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].
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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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 25 (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
The Classification of Semisimple Hopf Algebras of dimension 16
 J. of Algebra
"... Abstract. In this paper we completely classify nontrivial semisimple Hopf algebras of dimension 16. We also compute all the possible structures of the Grothendieck ring of semisimple noncommutative Hopf algebras of dimension 16. Moreover, we prove that noncommutative semisimple Hopf algebras of di ..."
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Cited by 24 (1 self)
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Abstract. In this paper we completely classify nontrivial semisimple Hopf algebras of dimension 16. We also compute all the possible structures of the Grothendieck ring of semisimple noncommutative Hopf algebras of dimension 16. Moreover, we prove that noncommutative semisimple Hopf algebras of dimension p n, p is prime, cannot have a cyclic group of grouplikes. 1. Introduction. Recently various classification results were obtained for finitedimensional semisimple Hopf algebras over an algebraically closed field of characteristic 0. The smallest dimension, for which the question was still open, was 16. In this paper we completely classify all nontrivial (i.e. noncommutative and noncocommutative) Hopf algebras of dimension 16. Moreover, we consider all
Finite quantum groups over abelian groups of prime exponent
"... Since the discovery of quantum groups (Drinfeld, Jimbo) and finite dimensional variations thereof (Lusztig, Manin), these objects were studied from different points of view and had many applications. The present paper is part of a series where we intend to show that important classes of Hopf algebra ..."
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Cited by 22 (3 self)
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Since the discovery of quantum groups (Drinfeld, Jimbo) and finite dimensional variations thereof (Lusztig, Manin), these objects were studied from different points of view and had many applications. The present paper is part of a series where we intend to show that important classes of Hopf algebras are quantum groups and therefore belong to Lie theory. One of our main results is the explicit
Semisimple Hopf algebras of dimension pq are trivial
 Journal of Algebra
, 1998
"... This paper makes a contribution to the problem of classifying finitedimensional semisimple Hopf algebras H over an algebraically closed field k of characteristic 0. Specifically, we show that if H has dimension pq for primes p and q then H is trivial, that is, H is either a group algebra or the dua ..."
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Cited by 20 (3 self)
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This paper makes a contribution to the problem of classifying finitedimensional semisimple Hopf algebras H over an algebraically closed field k of characteristic 0. Specifically, we show that if H has dimension pq for primes p and q then H is trivial, that is, H is either a group algebra or the dual of a group algebra. Previously known cases include dimension 2p [M1], dimension p 2 [M2], and dimensions 3p, 5p and 7p [GW]. Westreich and the second author also obtained the same result for H which is, along with its dual H ∗ , of Frobenius type (i.e. the dimensions of their irreducible representations divide the dimension of H) [GW, Theorem 3.5]. They concluded with the conjecture that any semisimple Hopf algebra H of dimension pq over k is trivial. In this paper we use Theorem 1.4 in [EG] to prove that both H and H ∗ are of Frobenius type, and hence prove this conjecture. Throughout k will denote an algebraically closed field of characteristic 0, and p and q will denote two prime numbers satisfying p < q. We let G(H) denote the group of grouplike elements of a Hopf algebra H, and D(H) denote its Drinfel’d double. Recall that H and H ∗cop are Hopf subalgebras of D(H), and let iH: H ֒ → D(H) and iH ∗cop: H∗cop ֒ → D(H)