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inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a by-product we obtain results about the Frobenius-Schur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the description of boundary conditions in twodimensional conformal field theory and present illustrative examples. We show that when the module category is tensor, then it gives rise to a NIM-rep of the fusion rules, and discuss a possible relation with the representation theory of vertex operator algebras. 1 CFT boundary conditions Boundary conditions in conformal field theory have various physical applications, ranging from the study of defects in condensed matter physics to the theory of open strings. Such boundary conditions are partially characterized by the maximal vertex operator subalgebra A of the bulk chiral algebra Abulk that they respect [43, 75]. That A is respected by a boundary condition means that the

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

If A is a weak C∗-Hopf algebra then the category of finite dimensional unitary representations of A is a monoidal C∗-category with monoidal unit being the GNS representation Dε associated to the counit ε. This category has isomorphic left dual and right dual objects which leads, as usual, to the notion of dimension function. However, if ε is not pure the dimension function is matrix valued with rows and columns labelled by the irreducibles contained in Dε. This happens precisely when the inclusions AL ⊂ A and AR ⊂ A are not connected. Still there exists a trace on A which is the Markov trace for both inclusions. We derive two numerical invariants for each C∗-WHA of trivial hypercenter. These are the common indices I and δ, of the Haar, respectively Markov conditional expectations of either one of the inclusions AL/R ⊂ A and ÂL/R ⊂ Â. In generic cases I> δ. In the special case of weak Kac algebras we show that I = δ is an integer. Submitted to J. Algebra

Let H be a finite-dimensional semisimple Hopf algebra. Recently it was shown in [LM] that a version of the Frobenius-Schur theorem holds for Hopf algebras, and thus that the Schur indicator ν(χ) of the character χ of a simple H-module is well-defined; this fact for the special case of Kac algebras was shown in [FGSV]. In this paper we

Abstract. In this paper, we define the higher Frobenius-Schur (FS-)indicators for finite-dimensional modules V of a semisimple quasi-Hopf algebra H via the categorical counterpart developed in [NS]. We prove that this definition of higher FS-indicators coincides with the higher indicators introduced by Kashina, Sommerhäuser, and Zhu when H is a Hopf algebra. We also obtain a sequence of canonical central elements of H, which is invariant under gauge transformations, whose values, when evaluated by the character of an H-module V, are the higher Frobenius-Schur indicators of V. As an application, we show that FS-indicators are sufficient to distinguish the four gauge classes of semisimple quasi-Hopf algebras of dimension eight corresponding to the four fusion categories with certain fusion rules classified by Tambara and Yamagami. Three of these categories correspond to well-known Hopf algebras, and we explicitly construct a quasi-Hopf algebra corresponding to the fourth. We also derive the formula of FS-indicators for the twisted quantum doubles of finite groups.

Abstract. We obtain two formulae for the higher Frobenius-Schur indicators: one for a spherical fusion category in terms of the twist of its center and the other one for a modular tensor category in terms of its twist. The first one is a categorical generalization of an analogous result by Kashina, Sommerhäuser, and Zhu for Hopf algebras, and the second one extends Bantay’s 2nd indicator formula for a conformal field theory to higher degree. These formulae imply the sequence of higher indicators of an object in these categories is periodic. We define the notion of Frobenius-Schur (FS-)exponent of a pivotal category to be the global period of all these sequences of higher indicators, and we prove that the FS-exponent of a spherical fusion category is equal to the order of the twist of its center. Consequently, the FS-exponent of a spherical fusion category is a multiple of its exponent by a factor not greater than 2. As applications of these results, we prove that the FS-exponent of a semisimple quasi-Hopf algebra H has the same set of prime divisors as of dim(H) and it divides dim(H) 4. In addition, if H is a group-theoretic quasi-Hopf algebra, the FS-exponent of H divides dim(H) 2, and this upper bound is shown to be tight. 1.

Abstract. We define higher Frobenius-Schur indicators for objects in linear pivotal monoidal categories. We prove that they are category invariants, and take values in the cyclotomic integers. We also define a family of natural endomorphisms of the identity endofunctor on a k-linear semisimple rigid monoidal category, which we call the Frobenius-Schur endomorphisms. For a k-linear semisimple pivotal monoidal category — where both notions are defined —, the Frobenius-Schur indicators can be computed as traces of the Frobenius-Schur endomorphisms.

Abstract. Mason and Ng have given a generalization to semisimple quasi-Hopf algebras of Linchenko and Montgomery’s generalization to semisimple Hopf algebras of the classical Frobenius-Schur theorem for group representations. We give a simplified proof, in particular a somewhat conceptual derivation of the appropriate form of the Frobenius-Schur indicator that indicates if and in which of two possible fashions a given simple module is self-dual. 1.

In this paper we study the representations of two semisimple Hopf algebras related to the symmetric group Sn, namely the bismash products Hn = k Cn #kSn−1 and its dual Jn = k Sn−1 #kCn = (Hn) ∗ , where k is an algebraically closed field of