We analyze the induction and restriction of sectors for nets of subfactors defined by Longo and Rehren. Picking a local subfactor we derive a formula which specifies the structure of the induced sectors in terms of the original DHR sectors of the smaller net and canonical endomorphisms. We also obtain a reciprocity formula for induction and restriction of sectors, and we prove a certain homomorphism property of the induction mapping. Developing further some ideas of F. Xu we will apply this theory in a forthcoming paper to nets of subfactors arising from conformal field theory, in particular those coming from conformal embeddings or orbifold inclusions of SU(n) WZW models. This will provide a better understanding of the labeling of modular invariants by certain graphs, in particular of the A-D-E classification of SU(2) modular invariants.

Dedicated to John E. Roberts on the occasion of his sixtieth birthday We describe the structure of the inclusions of factors A(E) ⊂A(E ′ ) ′ associated with multi-intervals E ⊂ R for a local irreducible net A of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion associated with two separated intervals is isomorphic to the Longo-Rehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of A. As a consequence, the index of A(E) ⊂A(E ′ ) ′ coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is non-degenerate, namely the representations of A form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry.

In this paper we further analyze modular invariants for subfactors, in particular the structure of the chiral induced systems of M-M morphisms. The relative braiding between the chiral systems restricts to a proper braiding on their “ambichiral ” intersection, and we show that the ambichiral braiding is non-degenerate if the original braiding of the N-N morphisms is. Moreover, in this case the dimensions of the irreducible representations of the chiral fusion rule algebras are given by the chiral branching coefficients which describe the ambichiral contribution in the irreducible decomposition of α-induced sectors. We show that modular invariants come along naturally with several nonnegative integer valued matrix representations of the original N-N Verlinde fusion rule algebra, and we completely determine their decomposition into its characters. Finally the theory is illustrated by various examples, including the treatment of all SU (2)k modular invariants.

Abstract. The goal of this paper is to classify “finite subgroups in Uq(sl2)” where q = e πi/l is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of Uq(sl2); we show that this definition is a natural generalization of the notion of a subgroup in a reductive group, and that it is also related with extensions of the chiral (vertex operator) algebra corresponding to ̂ sl2 at level k = l − 2. We show that “finite subgroups in Uq(sl2) ” are classified by Dynkin diagrams of types An, D2n, E6, E8 with Coxeter number equal to l, give a description of this correspondence similar to the classical McKay correspondence, and discuss relation with modular invariants in ( ̂ sl2)k conformal field theory.

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

We consider a type III subfactor N ⊂ M of finite index with a finite system of braided N-N morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply α-induction and, developing further some ideas of Ocneanu, we define chiral generators for the double triangle algebra. Using a new concept of intertwining braiding fusion relations, we show that the chiral generators can be naturally identified with the α-induced sectors. A matrix Z is defined and shown to commute with the S- and Tmatrices arising from the braiding. If the braiding is non-degenerate, then Z is a “modular invariant mass matrix ” in the usual sense of conformal field theory. We show that in that case the fusion rule algebra of the dual system of M-M morphisms is generated by the images of both kinds of α-induction, and that the structural information about its irreducible representations is encoded in the mass matrix Z. Our analysis sheds further light on the connection between (the classifications of) modular invariants and subfactors, and we will

We investigate the structure of the Longo-Rehren inclusion for a finite closed system of endomorphisms of factors, whose categorical structure is known to be the same as the asymptotic inclusion of A. Ocneanu. In particular, we obtain a precise description of the sectors associated with the Longo-Rehren inclusions in terms of half braidings, which do not necessarily satisfy the usual condition of braidings. In doing so, we give new proofs to most of the known statements concerning asymptotic inclusions. We construct a complete system of matrix units of the tube algebra using the half braidings, which will be used in the second part to describe concrete examples of the Longo-Rehren inclusions arising from the Cuntz algebra endomorphisms. We also discuss the case where the original system has a braiding, and generalize Ocneanu and Evans-Kawahigashi's method for the analysis of the asymptotic inclusions of the Hecke algebra subfactors. 1 Introduction The notion of the asymptotic inclusio...

A braided subfactor determines a coupling matrix Z which commutes with the S- and T-matrices arising from the braiding. Such a coupling matrix is not necessarily of “type I”, i.e. in general it does not have a block-diagonal structure which can be reinterpreted as the diagonal coupling matrix with respect to a suitable extension. We show that there are always two intermediate subfactors which correspond to left and right maximal extensions and which determine “parent ” coupling matrices Z ± of type I. Moreover it is shown that if the intermediate subfactors coincide, so that Z + = Z − , then Z is related to Z + by an automorphism of the extended fusion rules. The intertwining relations of chiral branching coefficients between original and extended S- and T-matrices are also clarified. None of our results depends on non-degeneracy of the braiding, i.e. the S- and T-matrices need not be modular. Examples from SO(n) current algebra models illustrate that the parents can be different, Z + ̸ = Z − , and that Z need not be related to a type I invariant by such an automorphism. 1

Abstract. Some mathematical questions relating to Coset Conformal field theories (CFT) are considered in the framework of Algebraic Quantum Field Theory as developed previously by us. We consider the issue of fix point resolution in the diagonal coset of type A. We show how to decompose certain reducible representations into irreducibles, and prove that the coset CFT gives rise to a unitary Modular Tensor Category in the sense of Turaev, and therefore may be used to construct 3-manifold invariants. We prove that if the coset inclusion satisfies certain conditions which can be checked in examples, the Kac-Wakimoto Hypothesis (KWH) is equivalent to the the Kac-Wakimoto Conjecture (KWC), a result which seems to be hard to prove by purely representation considerations. Examples are also presented. This paper is a sequel to [X4]. Let us first recall some definitions from [X4]. Let G be a simply connected compact Lie group and let H ⊂ G be a Lie subgroup. Let π i be an irreducible representations of LG with positive energy at level k 1 on Hilbert space H i. Suppose when restricting to LH, H i decomposes as: