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

We characterize finite index depth 2 inclusions of type II1 factors in terms of actions of weak Kac algebras and weak C ∗-Hopf algebras. If N ⊂ M ⊂ M1 ⊂ M2 ⊂... is the Jones tower constructed from such an inclusion N ⊂ M, then B = M ′ ∩ M2 has a natural structure of a weak Kac algebra (if the index [M: N] is integer) or a weak C ∗-Hopf algebra (if [M: N] is non-integer) and there is a minimal action of B on M1 such that M is the fixed point subalgebra of M1 and M2 is isomorphic to the crossed product of M1 and B. This extends the well-known results for irreducible depth 2 inclusions. 1

Let (A; 1) be a finite dimensional unital associative algebra over a field K, which is also equipped with a coassociative counital coalgebra structure (\Delta; "). A is called a weak bialgebra if the coproduct \Delta : A ! A\Omega A satisfies \Delta(ab) = \Delta(a)\Delta(b). We do not require \Delta(1) = 1\Omega 1 nor multiplicativity of the counit " : A ! K. Instead, we propose a new set of counit axioms, which are modelled so as to guarantee that RepA becomes a monoidal category with unit object given by the cyclic left A-module E := (A * 1) ae A, where 1 j " is the unit in the dual weak bialgebra A. Under these monoidality axioms E and ¯ E := ( 1 ( A) become commuting unital subalgebras of A, which are trivial if and only if " is multiplicative. We also propose axioms for an antipode S : A ! A, such that the category RepA becomes rigid. S is uniquely determined, provided it exists. If a monoidal weak bialgebra A has an antipode S, then its dual A is monoidal if and only...

Abstract. We bring together ideas in analysis of Hopf ∗-algebra actions on II1 subfactors of finite Jones index [9, 24] and algebraic characterizations of Frobenius, Galois and cleft Hopf extensions [14, 13, 3] to prove a non-commutative algebraic analogue of the classical theorem: a finite field extension is Galois iff it is separable and normal. Suppose N ֒ → M is a separable Frobenius extension of k-algebras split as N-bimodules with a trivial centralizer CM(N). Let M1: = End(MN) and M2: = End(M1)M be the endomorphism algebras in the Jones tower N ֒ → M ֒ → M1 ֒ → M2. We show that under depth 2 conditions on the second centralizers A: = CM1 (N) and B: = CM2 (M) the algebras A and B are semisimple Hopf algebras dual to one another and such that M1 is a smash product of M and A, and that M is a B-Galois extension of N. 1.

Starting from a local quantum field theory with an unbroken compact symmetry group G in 1 + 1-dimensional spacetime we construct disorder fields implementing gauge transformations on the fields (order variables) localized in a wedge region. Enlarging the local algebras by these disorder fields we obtain a nonlocal field theory, the fixpoint algebras of which under the appropriately extended action of the group G are shown to satisfy Haag duality in every simple sector. The specifically 1+1 dimensional phenomenon of violation of Haag duality of fixpoint nets is thereby clarified. In the case of a finite group G the extended theory is acted upon in a completely canonical way by the quantum double D(G) and satisfies R-matrix commutation relations as well as a Verlinde algebra. Furthermore, our methods are suitable for a concise and transparent approach to bosonization. The main technical ingredient is a strengthened version of the split property which should hold in all reasonable massive theories. In the appendices (part of) the results are extended to arbitrary locally compact groups and our methods are adapted to chiral theories on the circle. 1

We apply the theory of finite dimensional weak C*-Hopf algebras A as developed by G. Bohm, F. Nill and K. Szlachányi [BSz,Sz,N1,N2,BNS] to study reducible inclusion triples of von-Neumann algebras N ae M ae M ? / A, where M is an A-module algebra with left A-action . : A\Omega M ! M, N j M A is the fixed point algebra and M ? / A is the crossed product extension. Here "weak" means that the coproduct \Delta on A is non-unital, requiring various modifications of the standard definitions for Hopf (co-)actions and crossed products. We show that normalized positive and nondegenerate left integrals l 2 A give rise to faithful conditional expectations E l : M ! N via E l (m) := l . m, where under certain regularity conditions this correspondence is one-to-one. Associated with such left integrals we construct "Jones projections" e l 2 A obeying for all m 2 M the Jones relations e l me l = E l (m)e l = e l E l (m) as an identity in M ? / A. We also present a concept of Plancherel...

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

We establish the equivalence of three versions of a finite dimensional quantum groupoid: a generalized Kac algebra introduced by T. Yamanouchi, a weak C ∗-Hopf algebra introduced by G. Böhm– F. Nill–K. Szlachányi (with an involutive antipode), and a Kac bimodule – an algebraic version of a Hopf bimodule, the notion introduced by J.-M. Vallin. We also study the structure and construct examples of finite dimensional quantum groupoids.

Abstract. We study a symmetric Markov extension of k-algebras N ֒ → M, a certain kind of Frobenius extension with conditional expectation that is tracial on the centralizer and dual bases with a separability property. We place a depth two condition on this extension, which is essentially the requirement that the Jones tower N ֒ → M ֒ → M1 ֒ → M2 can be obtained by taking relative tensor products with centralizers A = CM1 (N) and B = CM2 (M). Under this condition, we prove that N ֒ → M is the invariant subalgebra pair of a weak Hopf algebra action by A, i.e., that N = MA. The endomorphism algebra M1 = EndNM is shown to be isomorphic to the smash product algebra M#A. We also extend results of Szymański [26], Vainerman and the second author [18], and the authors [11].

Abstract. We consider a condition for non-degenerate commuting squares of matrix algebras (finite dimensional von Neumann algebras) called the span condition, which in the case of the n-dimensional standard spin models is shown to be satisfied if and only if n is prime. We prove that the commuting squares satisfying the span condition are isolated among all commuting squares (modulo isomorphisms). In particular, they are finiteley many for any fixed dimension. Also, we give a conceptual proof of previous constructions of certain one-parameter families of biunitaries. 1.