## From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories (2003)

Venue: | J. Pure Appl. Alg |

Citations: | 54 - 7 self |

### BibTeX

@ARTICLE{Müger03fromsubfactors,

author = {Michael Müger},

title = {From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories},

journal = {J. Pure Appl. Alg},

year = {2003}

}

### OpenURL

### Abstract

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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Citation Context ...alent to the weaker condition of invertibility of ∑ i xiyi (proven in [1] for the commutative and in [58] for the symmetric case). Thus canonical Frobenius algebras are semisimple. ✷ It is well known =-=[38]-=- that every finite dimensional Hopf algebra over a field F is a Frobenius algebra. Our aim in the remainder of this subsection is to clarify when these Frobenius algebras are canonical. We recall some... |

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Citation Context ...ruction. (Module categories automatically have subobjects.) To prove this rigorously it is, in view of Theorem 3.17, sufficient to prove that ˜ E satisfies the requirements of the latter. Inspired by =-=[18]-=- (which in turn was influenced by ideas of the author), the alternative definition of ˜ E was proposed also by S. Yamagami in [76], which he kindly sent me. 2. By the above remark, our category B = EN... |

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Citation Context ...ave BW(M, A) = BW(M, B) for all closed orientable 3-manifolds M. Remark 7.2 1. Before we sketch the proof of this result we point out that it resolves a (minor) puzzle concerning the BW invariant. In =-=[36]-=- Kuperberg had defined a 3-manifold invariant Ku(M, H) for every finite dimensional Hopf algebra H over an algebraically closed field F which is involutive (S 2 = id) and whose characteristic does not... |

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Citation Context ...logy. Braided tensor categories have in fact served as an input in new constructions of invariants of links and 3-manifolds and of topological quantum field theories [69, 31]. (Recently it turned out =-=[3, 20]-=- that a braiding is not needed for the construction of the triangulation or ‘state sum’ invariant of 3-manifolds.) A particular rôle in this context has been played by subfactor theory, see e.g. [24, ... |

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Citation Context ...work of Longo [39, 40]. Those aspects of subfactor theory [40, 23] which most directly inspired the present investigation and [47] were in fact done in the type III setting. Anyway, by Popa’s results =-=[57]-=- the classifications of amenable inclusions of hyperfinite type II1 and III1 factors amount to the same thing. The following is implicit in much of the literature on type III subfactors and explicit i... |

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on Axiomatic Topological Quantum Field Theory, LAS/Park City
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Citation Context ...near form φ : A → F for which the bilinear form b(a,b) = φ(ab) is non-degenerate. (For a nice exposition of the present state of Frobenius theory we refer to [29].) Recent results of Quinn and Abrams =-=[58, 1, 2]-=- provide the following alternative characterization: A Frobenius algebra is a quintuple (A, m,η, ∆, ε), where (A, m, η) and (A,∆, ε) are a finite dimensional algebra and coalgebra, respectively, over ... |

37 | The structure of sectors associated with Longo-Rehren inclusions
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Citation Context ...atter seems to have lost some of the attention of the wider public. This is deplorable, since operator algebraists continue to generate ideas whose pertinence extends beyond subfactor theory, e.g. in =-=[53, 15, 74, 23]-=-. The ∗ AMS subject classification: 18D10, 18D05; 46L37 † Supported by EU through the TMR Networks “Noncommutative Geometry” and “Algebraic Lie Representations”, by MSRI through NSF grant DMS-9701755 ... |

37 |
tensor categories from quantum groups
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Citation Context ...d dimension with the expected properties. 2.4 Duality in ∗-Categories In this section we posit F = C. Many complex linear categories have an additional piece of structure: a positive ∗-operation. See =-=[12, 71]-=- for two important classes of examples. A ∗-operation on a C-linear category is map which assigns to every morphism s : X → Y a morphism s ∗ : Y → X. This map has to be antilinear, involutive (s ∗∗ = ... |

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Citation Context ...ism). Yet there are tensor categories where every object has a two-sided dual, in particular rigid symmetric categories (=closed categories [44]), rigid braided ribbon categories (=tortile categories =-=[27]-=-), ∗-categories [42], and most generally, pivotal categories. (Functors, i.e. 1-morphisms in CAT , which have a two-sided adjoint are occasionally called ‘Frobenius functors’.) If a 1-morphism F happe... |

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Citation Context ...on n of the conjugate equations for X then d(X) ≤ r ∗ ◦ r and equality holds iff (X, r,r) is standard. For the proofs we refer to [42]. Furthermore, a braided ∗-category with conjugates automatically =-=[46]-=- has a canonical twist [28]. Together with the fact [78] that in braided tensor categories there is a one-to-one correspondence between sovereign structures and twists this implies that every braided ... |

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29 |
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Citation Context ...(M,H − mod) = BW(M, ˆ H − mod) can be derived from the connection between the invariants BW and Ku. A less obvious example is provided in the companion paper [47]. There we prove that the center Z(C) =-=[45, 27, 62]-=-, which is the categorical version of Drinfel’d’s quantum double, of a semisimple spherical category with non-zero dimension is again spherical and semisimple (and modular in the sense of Turaev). Fur... |

28 | From subfactors to categories and topology III, in preparation
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(Show Context)
Citation Context ... that there is a one-to-one correspondence between twists and spherical structures, cf. [78].) We believe that this is the most general setting within which results like Proposition 5.17 and those of =-=[47]-=- obtain without a fundamental change of the methods. (∗-categories can be turned into spherical categories, cf. [75], but this is not necessarily the most convenient thing to do.) The following result... |

28 | Finite index subfactors and Hopf algebra crossed products - Szymanski - 1994 |

25 | Spherical categories
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Citation Context ...resentations, and therefore dim Hom(X, Ql) = d(X) for all simple X. Furthermore, the regular representation is absorbing: X ⊗ Ql ∼ = Ql ⊗ X ∼ = d(X)Ql for every X ∈ A − mod. Proof of Theorem 6.15. By =-=[4]-=-, the category H − mod is spherical and by the coherence theorem [4] we may consider H − mod as strict monoidal and strict spherical. By Theorem 6.12 we have a canonical and irreducible Frobenius alge... |

24 |
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Citation Context ...ich suggests connections with topology. Braided tensor categories have in fact served as an input in new constructions of invariants of links and 3-manifolds and of topological quantum field theories =-=[69, 31]-=-. (Recently it turned out [3, 20] that a braiding is not needed for the construction of the triangulation or ‘state sum’ invariant of 3-manifolds.) A particular rôle in this context has been played by... |

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Citation Context ...and cosemisimple and (6.2) coincides with dim H · 1F. 40Proof. Eq. (6.2) follows from the above computation of v ′ ◦ v and w ′ ◦ w. If (6.2) is non-zero then H is semisimple and cosemisimple, and by =-=[14]-=- the antipode is involutive. By [37, Theorem 2.5], (6.2) coincides with tr(S 2 ) and therefore with dim H · 1F. � By the preceding result semisimple and cosemisimple Hopf algebras provide examples of ... |

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Citation Context ... category theory and subfactor theory. 1.2 Adjoint Functors and Adjoint Morphisms We assume the reader to be conversant with the basic definitions of categories, functors and natural transformations, =-=[44]-=- being our standard reference. (In the next section we will recall some of the relevant definitions.) As is well known, the concept of adjoint functors is one of the most important ones not only in ca... |

18 |
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Citation Context ...atter seems to have lost some of the attention of the wider public. This is deplorable, since operator algebraists continue to generate ideas whose pertinence extends beyond subfactor theory, e.g. in =-=[53, 15, 74, 23]-=-. The ∗ AMS subject classification: 18D10, 18D05; 46L37 † Supported by EU through the TMR Networks “Noncommutative Geometry” and “Algebraic Lie Representations”, by MSRI through NSF grant DMS-9701755 ... |

18 |
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(Show Context)
Citation Context ...d category minded people, the only new result being Theorem 6.28. We begin with a very brief definition of the notions we will use. For everything else see any textbook on von Neumann algebras, e.g., =-=[64, 68, 60]-=- and subfactors [26, 16]. A von Neumann algebra (vNa) is a unital subalgebra M ⊂ B(H) of the algebra B(H) of bounded operators on some Hilbert space H which is closed w.r.t. the hermitian conjugation ... |

16 |
Modules, comodules and cotensor products over Frobenius algebras
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Citation Context ...er authors. We summarize these references and comment on several other recent works. The relation between classical Frobenius algebras and Frobenius algebras in F-Vect is due to Quinn [58] and Abrams =-=[1, 2]-=-. The literature on Frobenius algebras in categories other than Vect is quite small but has begun to grow recently. As mentioned earlier, canonical Frobenius algebras in C ∗ -categories (‘Q-systems’) ... |

16 |
Subfactors and knots
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(Show Context)
Citation Context ...m’ invariant of 3-manifolds.) A particular rôle in this context has been played by subfactor theory, see e.g. [24, 52, 26, 16], which has led to the discovery of Jones’ polynomial invariant for knots =-=[25]-=-. Since the Jones polynomial was quickly reformulated in more elementary terms, and due to the technical difficulty of subfactor theory, the latter seems to have lost some of the attention of the wide... |

14 | The equality of 3-manifold invariants
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(Show Context)
Citation Context ...closed field F which is involutive (S 2 = id) and whose characteristic does not divide the dimension of H. (That these conditions are equivalent to semisimplicity of H and ˆ H was not known then.) In =-=[5]-=- it was proven that the invariant BW is a generalization of Ku in the sense that Ku(M, H) = BW(M,H − mod), again assuming appropriate normalizations. For Kuperberg’s invariant it had been known that K... |