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

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Abstract. We generalize the fundamental structure Theorem on Hopf (bi)modules by Larson and Sweedler to quasi-Hopf algebras H. For dim H < ∞ this proves the existence and uniqueness (up to scalar multiples) of integrals in H. Among other applications we prove a Maschke-type Theorem for diagonal crossed products as constructed by the authors in [HN, HN99].

We study the general structure of 2-C ∗-categories closed under conjugation, projections and direct sums. We do not assume units to be simple, i.e. for ιA the 1-unit corresponding to an object A, the space Hom(ιA, ιA) is a commutative unital C ∗-algebra. We show that 2-arrows can be viewed as continuous sections in Hilbert bundles and describe the behaviour of the fibres with respect to the categorical structure. We give an example of a 2-C ∗-Category giving rise to bundles of finite Hopf-algebras in duality. We make some remarks concerning Frobenius algebras and Q-systems in the general context of tensor C ∗-categories with non-simple units.

Integrals play a basic role in the structure theory of finite-dimensional Hopf algebras A and their duals A ∗ over a field k. The existence and uniqueness of integrals for finite-dimensional Hopf algebras was first established by Hopf modules; for a long time the only means known for doing so. In [2], [3] and [7] the existence and uniqueness of integrals for A is established without using Hopf modules. We find the approach of [3] very interesting in that it is based on a formalism which relates Hopf algebras and complete invariants of 3-manifolds in a rather intriguing way. This paper has two main purposes. The first is to explain the formalism of [3] to the extent that the proof of the existence and uniqueness of integrals for A found in [3] can be understood in more familiar algebraic terms. The reader is directed to [10] for a much fuller explanation of this formalism which includes a discussion of its subtleties and its connections with identities which

and some examples of quantum groupoids and vertex groups related to them

We show for any oriented surface, possibly with a boundary, how to generalize Kramers–Wannier duality to the world of quantum groups. The generalization is motivated by quantization of Poisson–Lie T-duality from the string theory. Cohomologies with quantum coefficients are defined for surfaces and their meaning is revealed. They are functorial with respect to some glueing operations and connected with q-invariants of 3-folds. 1