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13
From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories
 J. Pure Appl. Alg
, 2003
"... 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 f ..."
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Cited by 52 (6 self)
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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 = FVect, where F is a field. An object X ∈ A with twosided 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 3manifolds 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
On quantum Galois theory
 Duke Math. J
, 1997
"... The goals of the present paper are to initiate a program to systematically study and rigorously establish what a physicist might refer to as the “operator content of orbifold models. ” To explain what this might mean, and to clarify the title of the paper, we will assume that the reader is familiar ..."
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Cited by 43 (14 self)
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The goals of the present paper are to initiate a program to systematically study and rigorously establish what a physicist might refer to as the “operator content of orbifold models. ” To explain what this might mean, and to clarify the title of the paper, we will assume that the reader is familiar with the algebraic formulation of 2dimensional
Representations of compact quantum groups and subfactors
 J. Reine Angew. Math
, 1999
"... Abstract: We associate Popa systems ( = standard invariants of subfactors, cf. [P3],[P4]) to the finite dimensional representations of compact quantum groups. We characterise the systems arising in this way: these are the ones which can be “represented ” on finite dimensional Hilbert spaces. This is ..."
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Cited by 32 (18 self)
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Abstract: We associate Popa systems ( = standard invariants of subfactors, cf. [P3],[P4]) to the finite dimensional representations of compact quantum groups. We characterise the systems arising in this way: these are the ones which can be “represented ” on finite dimensional Hilbert spaces. This is proved by an universal construction. We explicitely compute (in terms of some free products) the operation of going from representations of compact quantum groups to Popa systems and then back via the universal construction. This is related with our previous work [B2]. We prove a Kesten type result for the coamenability of compact quantum groups, which allows us to compare it with the amenability of subfactors.
Limit theorems and absorption problems for quantum random walks
 in one dimension, Quantum Information and Computation
, 2002
"... Abstract. In this paper we consider limit theorems, symmetry of distribution, and absorption problems for two types of onedimensional quantum random walks determined by 2 × 2 unitary matrices using our PQRS method. The one type was introduced by Gudder in 1988, and the other type was studied intens ..."
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Cited by 7 (4 self)
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Abstract. In this paper we consider limit theorems, symmetry of distribution, and absorption problems for two types of onedimensional quantum random walks determined by 2 × 2 unitary matrices using our PQRS method. The one type was introduced by Gudder in 1988, and the other type was studied intensively by Ambainis et al. in 2001. The difference between both types of quantum random walks is also clarified. 1
Approximately finitely acting operator algebras
 Department of Mathematics and Statistics, University of Guelph
"... Abstract. Let E be an operator algebra on a Hilbert space with finitedimensional C*algebra C ∗ (E). A classification is given of the locally finite algebras A0 = alg lim(Ak, φk) and the operator algebras A = lim(Ak, φk) obtained as limits of direct sums of matrix algebras over E with respect to sta ..."
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Cited by 4 (2 self)
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Abstract. Let E be an operator algebra on a Hilbert space with finitedimensional C*algebra C ∗ (E). A classification is given of the locally finite algebras A0 = alg lim(Ak, φk) and the operator algebras A = lim(Ak, φk) obtained as limits of direct sums of matrix algebras over E with respect to starextendible homomorphisms. The invariants in the algebraic case consist of an additive semigroup, with scale, which is a right module for the semiring VE = Homu(E ⊗K, E ⊗K) of unitary equivalence classes of starextendible homomorphisms. This semigroup is referred to as the dimension module invariant. In the operator algebra case the invariants consist of a metrized additive semigroup with scale and a contractive right module VEaction. Subcategories of algebras determined by restricted classes of embeddings, such as 1decomposable embeddings between digraph
Cohomology For Finite Index Inclusions Of Factors
"... If N ` M is an inclusion of type II 1 factors of finite index on a separable Hilbert space, and if N has a Cartan subalgebra then we show that H n (N ; M) = 0 for n 1. We also show that H n cb (N ; M) = 0, n 1, for an arbitrary finite index inclusion N ` M of von Neumann algebras. 1 Introducti ..."
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Cited by 3 (3 self)
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If N ` M is an inclusion of type II 1 factors of finite index on a separable Hilbert space, and if N has a Cartan subalgebra then we show that H n (N ; M) = 0 for n 1. We also show that H n cb (N ; M) = 0, n 1, for an arbitrary finite index inclusion N ` M of von Neumann algebras. 1 Introduction The continuous Hochschild cohomology groups H n (N ; X ) for a von Neumann algebra N and a Banach Nbimodule X were first studied in a series of papers [J1], [J2], [JKR], [KR1], [KR2] by Johnson, Kadison and Partially supported by a grant from the National Science Foundation. Ringrose. The primary focus was on the case X = N . The KadisonSakai theorem on derivations, [K], [S], had established that H 1 (N ; N ) = 0 for all von Neumann algebras, and so it was natural to pose the question of whether H n (N ; N ) = 0 for all n 2. The work of [CES], [CS1], [CS2], [KR1] on completely bounded cohomology gave an affirmative answer in the cases of type I; II 1 and III von Neumann alg...
HOPF ∗ALGEBRAS ASSOCIATED TO BIUNITARY MATRICES
, 1999
"... There exist several results which associate subfactors to Hopf algebras and vice versa. The relation between these objects is still very unclear. On the other hand examples of subfactors may be constructed from certain combinatorial data, encoded in the socalled commuting squares. Thus one can hope ..."
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There exist several results which associate subfactors to Hopf algebras and vice versa. The relation between these objects is still very unclear. On the other hand examples of subfactors may be constructed from certain combinatorial data, encoded in the socalled commuting squares. Thus one can hope that new examples of Hopf algebras
HOPF ALGEBRAS AND BIUNITARY MATRICES
, 1999
"... There exist several constructions of subfactors using quantum groups and vice versa. The precise relation between this objects is far from being clear. On the other hand examples of subfactors may be constructed from certain combinatorial data, encoded in the socalled commuting squares. Thus one ca ..."
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There exist several constructions of subfactors using quantum groups and vice versa. The precise relation between this objects is far from being clear. On the other hand examples of subfactors may be constructed from certain combinatorial data, encoded in the socalled commuting squares. Thus one can hope that new examples of Hopf
Fusion of Positive Energy Representations of
"... and my parents for their biblical patiencePreface The contents of this dissertation are original, except where explicit reference is made to the work of others. The original material is the result of my own work and includes nothing which is the outcome of a collaborative effort. No part of this dis ..."
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and my parents for their biblical patiencePreface The contents of this dissertation are original, except where explicit reference is made to the work of others. The original material is the result of my own work and includes nothing which is the outcome of a collaborative effort. No part of this dissertation has been, or is currently being submitted for a degree, diploma or qualification at this or any other University. I am deeply grateful to my research supervisor, Dr. Antony Wassermann for suggesting the problems discussed in this thesis, his continual guidance and for sharing his ideas on conformal field theory and operator algebras. I would also like to thank Professor Graeme Segal for his interest in my work and many enlightening and stimulating conversations, Hans Wenzl for sending me some unpublished notes on braid group representations the use of which was central to the main computation of this thesis and Alex Selby, for his expertise on local systems at a time when I was struggling with the Knizhnik–Zamolodchikov equations. I am indebted in countless ways to Giovanni Forni and Alessio Corti. It is a pleasure to thank them both for their friendship, advice and much needed encouragement. Grazie.