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

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 60-th 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,

We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A = F-Vect, where F is a field. An object X ∈ A with two-sided dual X gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A ≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗-categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3-manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple Hopf algebras, for which we prove H − mod ≈ ˆH − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205. 1

Abstract. We study the group of group-like elements of a weak Hopf algebra and derive an analogue of Radford’s formula for the fourth power of the antipode S, which implies that the antipode has a finite order modulo a trivial automorphism. We find a sufficient condition in terms of Tr(S 2) for a weak Hopf algebra to be semisimple, discuss relation between semisimplicity and cosemisimplicity, and apply our results to show that a dynamical twisting deformation of a semisimple Hopf algebra is cosemisimple. 1.

Given a finite dimensional C -Hopf algebra H and its dual H we construct the infinite crossed product A = : : : ? / H ? / H ? / H ? / : : : and study its superselection sectors. A is the observable algebra of a generalized quantum spin chain with H-order and H-disorder symmetries, where by a duality transformation the role of order and disorder may also appear interchanged. If H = j CG is a group algebra then A becomes an ordinary G-spin model. We classify all DHR-sectors of A --- relative to some Haag dual vacuum representation --- and prove that their symmetry is described by the Drinfeld double D(H). To achieve this we construct localized coactions ae : A ! A\Omega D(H) and use a certain compressibility property to prove that they are universal amplimorphisms on A. In this way the double D(H) can be recovered from the observable algebra A as a universal cosymmetry. 1 E-mail: NILL@omega.physik.fu-berlin.de Supported by the DFG, SFB 288 "Differentialgeometrie und Quant...

We study the higher Frobenius-Schur 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.

In this paper we consider semisimple Hopf algebras of dimension pq over an algebraically closed field k of characteristic 0, where p and q are distinct prime numbers. Masuoka has proved that a semisimple Hopf algebra of dimension 2p over k, where p is an odd prime, is either a group algebra or a dual of a group algebra [Ma1]. The authors have pushed the analysis further and obtained the same result for semisimple A of dimension 3p, where p is a prime greater than 3 [GW]. Thus, a natural conjecture is: Conjecture 1: Any semisimple Hopf algebra of dimension pq over k, where p and q are distinct prime numbers, is either a group algebra or a dual of a group algebra. A well known property of A, a finite dimensional semisimple group algebra or a dual of a group algebra, is that it is of Frobenius type; that is, the dimension of any irreducible representation of A divides the dimension of A (the definition is due to Montgomery [Mo2]). A special case of one of Kaplansky’s conjectures [Kap] is: Conjecture 2: Any semisimple Hopf algebra of dimension pq over k, where p and q are distinct prime numbers, is of Frobenius type. In this paper we prove among the rest that Conjecture 1 is equivalent to Conjecture 2 (Theorem 2.7). A major role in the analysis is played by G(A), the group of grouplike elements of A. By [NZ], |G(A) | is either 1, p, q or pq. We prove in Theorem 2.1 that if p < q then |G(A) | ̸ = q, and if |G(A) | = p then q = 1(mod p). Consequently, we prove in Theorem 2.2 that if |G(A) | ̸ = 1 and q ̸ = 1(mod p) then A is a group algebra.

We prove that, over an algebraically closed field of characteristic zero, a semisimple Hopf algebra that has a nontrivial self-dual simple module must have even dimension. This generalizes a classical result of W. Burnside. As an application, we show under the same assumptions that a semisimple Hopf algebra that has a simple module of even dimension must itself have even dimension. 1 Suppose that H is a finite-dimensional Hopf algebra that is defined over the field K. We denote its comultiplication by ∆, its counit by ε, and its antipode by S. For the comultiplication, we use the sigma notation of R. G. Heyneman and M. E. Sweedler in the following variant: ∆(h) = h (1) ⊗ h (2) We view the dual space H ∗ as a Hopf algebra whose unit is the counit of H, whose counit is the evaluation at 1, whose antipode is the transpose of the antipode of H, and whose multiplication and comultiplication are determined

Conventions and Notation. 7

Abstract. The notion of kernel of a representation of a semisimple Hopf algebra is introduced. Similar properties to the kernel of a group representation are proved in some special cases. In particular, every normal Hopf subalgebra of a semisimple Hopf algebra H is the kernel of a representation of H. The maximal normal Hopf subalgebras of H are described.