## From subfactors to categories and topology III. Triangulation invariants of 3-manifolds and Morita equivalence of tensor categories

Venue: | In preparation |

Citations: | 27 - 3 self |

### BibTeX

@INPROCEEDINGS{Müger_fromsubfactors,

author = {Michael Müger},

title = {From subfactors to categories and topology III. Triangulation invariants of 3-manifolds and Morita equivalence of tensor categories},

booktitle = {In preparation},

year = {}

}

### OpenURL

### Abstract

### Citations

373 |
Quantum Groups
- Kassel
- 1995
(Show Context)
Citation Context ... algebra H is a certain Hopf algebra which contains H and the dual ˆ H as Hopf subalgebras and it is generated as an algebra by these. We refrain from repeating the well known definition and refer to =-=[22]-=- for a nice treatment. We only remark that D(H) ∼ = H ⊗F ˆ H as a vector space, thus dimF D(H) = (dimF H) 2 . Furthermore, D(H) is quasitriangular, i.e. there is an invertible R ∈ D(H) ⊗ D(H) such tha... |

271 |
Invariants of 3-manifolds via link polynomials and quantum
- Reshetikhin, Turaev
- 1991
(Show Context)
Citation Context ...3-manifolds There are two classes of invariants of 3-manifolds associated with a modular tensor category C, cf. [53]. On the one hand we have the surgery invariants RT(M, C) of Reshetikhin and Turaev =-=[50]-=- which are based on the fact that every connected oriented closed 3manifold can be obtained from S 3 by surgery along a framed link. It turned out [8] that modularity of the category C is not really n... |

89 | Module categories, weak Hopf algebras and modular invariants - Ostrik |

73 |
An associative orthogonal bilinear form for Hopf algebras
- Larson, Sweedler
- 1969
(Show Context)
Citation Context ...H,µ ∈ ˆ H which are traces in the sense that � 〈µ,ab〉 = 〈µ,ba〉, 〈αβ,Λ〉 = 〈βα,Λ〉 for all a,b ∈ H,αβ ∈ ˆ H. The category Rep D(H) is a spherical category. Proof. Semisimple Hopf algebras are unimodular =-=[28]-=-, which by definition means that there are two-sided integrals. By [28, Proposition 8], unimodular Hopf algebras satisfy 〈µ,ab〉 = 〈µ,bS 2 (a)〉 ∀a,b ∈ H, thus 〈µ, ·〉 is tracial by involutivity of S. Sp... |

69 |
A duality for Hopf algebras and for subfactors
- Longo
- 1994
(Show Context)
Citation Context ...irect sum γ = ⊕ i∈Γ ρi ⊗ ρ op i one shows [30] γ to be part of a Frobenius algebra (‘Q-system’) (Q,v,v ∗ ,w,w ∗ ) in Endf(A). At this point one applies a beautiful and fundamental result due to Longo =-=[29]-=-, which implies that there is a subfactor B ⊂ A such that γ is a canonical endomorphism for the inclusion B ⊂ A. This means that there is a normal morphism ι : A → B which is a dual (in the 2-category... |

68 |
Minimal quasitriangular Hopf algebras
- Radford
- 1993
(Show Context)
Citation Context ...4] for a detailed account. Now, the R-matrix of a quantum double D(H) is non-degenerate in a certain sense, D(H) being ‘factorizable’ [49]. If H is semisimple and cosemisimple then D(H) is semisimple =-=[47]-=-. It then turns out to be also modular and the category D(H)-mod of finite dimensional left D(H)-modules is modular in the sense of Turaev [53]. (This was proved in [13] 3for algebras over algebraica... |

63 | Multi-interval subfactors and modularity of representations in conformal field theory
- Kawahigashi, Longo, et al.
(Show Context)
Citation Context ... ◦ eY (Z) = eX(Z) ◦ s ∗ ⊗ idZ ∀Z, thus s ∗ ∈ Hom Z1(C)((Y,eY ),(X,eX)). � In the applications of the quantum double to operator algebras, like to the asymptotic subfactor [19] or quantum field theory =-=[25]-=-, one is mainly interested in the unitary quantum double. In order for the results of Theorem 1.2 to remain valid for Z ∗ 1 (C) ⊂ Z1(C) one must show Z ∗ 1 (C) that is equivalent to Z1(C) as a tensor ... |

62 |
Quantum R-matrices and factorization problems
- RESHETIKHIN
- 1988
(Show Context)
Citation Context ...-mod being provided by the R-matrix. Again, see [22, Chapter XIII.4] for a detailed account. Now, the R-matrix of a quantum double D(H) is non-degenerate in a certain sense, D(H) being ‘factorizable’ =-=[49]-=-. If H is semisimple and cosemisimple then D(H) is semisimple [47]. It then turns out to be also modular and the category D(H)-mod of finite dimensional left D(H)-modules is modular in the sense of Tu... |

61 |
A theory of dimension, K-theory 11
- Longo, Roberts
- 1997
(Show Context)
Citation Context ...ome once one realizes that the more algebraic part of subfactor theory can be cast into the language of 2-categories. This is the content of [38], which in a sense can be considered a continuation of =-=[31]-=-, though in a somewhat more general setting. The paper is organized as follows. In Section 2 we first recall some of the less standard definitions from [38]. We then summarize the main results of [38]... |

59 |
Braid group statistics and their superselection rules. In: The algebraic theory of superselection sectors. Introduction and recent results, Singapore: World Scientific
- Rehren
- 1990
(Show Context)
Citation Context ...g the properties in the theorem (i.e. a premodular category [8]) is modular iff the center Z2(C) is trivial, in the sense that all objects of Z2(C) are multiples of the tensor unit. (This was done in =-=[48]-=- for ∗-categories and in [7] for spherical categories with dim C ̸= 0, see also Corollary 7.11 below.) Thus Z2(Z1(C)) is trivial for all C as in the Main Theorem, which is the promised analogue of the... |

53 | Higher-dimensional algebra I: braided monoidal 2-categories. Available as ftp://math.ucr.edu/pub/baez/bm2cat.ps.Z
- Baez, Neuchl
(Show Context)
Citation Context ...logues to the above result in d = 1 and the trivial case d = 0 mentioned in the Introduction. (See [4] for a review of the theory of n-categories.) Thus, considering the center constructions in d = 2 =-=[3, 10]-=-, can one show that Z (2) 3 (Z(2) 2 (C)) is trivial? Here C, Z(2) 2 (C), Z(2) 3 (Z(2) 2 (C)) are (semisimple spherical) braided, sylleptic and symmetric 2-categories, respectively. ✷ 5.3 Computation o... |

52 | From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories
- Müger
(Show Context)
Citation Context ...ly relies on this structure.) Yet this problem can be overcome once one realizes that the more algebraic part of subfactor theory can be cast into the language of 2-categories. This is the content of =-=[38]-=-, which in a sense can be considered a continuation of [31], though in a somewhat more general setting. The paper is organized as follows. In Section 2 we first recall some of the less standard defini... |

51 | Rehren, Nets of subfactors
- Longo, K-H
- 1995
(Show Context)
Citation Context ...es not use category language and does not refer to the quantum double (center) of monoidal categories. In [15] Evans and Kawahigashi published proofs for most of Ocneanu’s announcements. In the paper =-=[30]-=-, which otherwise has little to do with the asymptotic subfactor, Longo and Rehren then constructed a subfactor B ⊂ A from an infinite factor M and a – in our language – finite dimensional full monoid... |

48 |
New Examples of Frobenius Extensions
- Kadison
- 1999
(Show Context)
Citation Context ...13 applies and B is semisimple. � Remark 3.15 Algebra extensions A ⊃ B admitting a conditional expectation E : A → B (satisfying certain conditions) are well known as Frobenius extensions, cf., e.g., =-=[21]-=- and are called Markov extensions if there is an E-invariant trace on A. ✷ Now we can put everything together: Theorem 3.16 Let F be algebraically closed and C a F-linear, spherical and semisimple ten... |

40 |
Invariants of piecewise-linear 3-manifolds, hep-th
- Barrett, Westbury
- 1996
(Show Context)
Citation Context ...ta equivalent. This is an equivalence relation which is considerably weaker than the usual equivalence, yet it implies that A and B have the same dimension and define the same triangulation invariant =-=[5, 16]-=- for 3-manifolds. See [38] for the details. 2. Unfortunately, the above statement of the theorem will not be sufficient for our purposes since beginning in Subsection 4.2 we will make use of the concr... |

37 |
Catégories prémodulaires, modularisations et invariants de variétés de dimension 3
- Bruguières
- 2000
(Show Context)
Citation Context ...ms, retractions, tensor products and duals, the inherited braiding obviously being symmetric. One can show that a braided category satisfying the properties in the theorem (i.e. a premodular category =-=[8]-=-) is modular iff the center Z2(C) is trivial, in the sense that all objects of Z2(C) are multiples of the tensor unit. (This was done in [48] for ∗-categories and in [7] for spherical categories with ... |

37 | The structure of sectors associated with the Longo-Rehren inclusions II
- Izumi
(Show Context)
Citation Context ...en in [37]. The author’s involvement in the present story began when in 1998 he received a copy of a short preprint [18] by M. Izumi. In the meantime a full account of Izumi’s results has appeared in =-=[19]-=-. In [18, 19] Izumi gives an in-depth analysis of the LR-subfactor, in particular its B − B sectors. Seeing [18] the present author was struck by the fact that its main theorem implicitly contained th... |

30 | Tortile Yang-Baxter operators in tensor categories - Joyal, Street - 1991 |

30 |
Braided groups and quantum Fourier transform
- Lyubashenko, Majid
- 1994
(Show Context)
Citation Context ...o [13]. A proof which covers also weak Hopf algebras (or finite quantum groupoids) is given in [41]. Our aim in this appendix is to give a proof which uses the ideas of Lyubashenko [32, 33] and Majid =-=[34]-=- and therefore is more in the spirit of our proof in the categorical 54situation. In the sequel H will always be a finite dimensional Hopf algebra. Since the main application will be to quantum doubl... |

26 |
Representations, duals and quantum doubles of monoidal categories
- Majid
- 1991
(Show Context)
Citation Context ...tion of Z0(X) to J. Baez.) Given an arbitrary monoidal category (or tensor category) C its center Z(C) is a braided monoidal category which was defined independently by Drinfel’d (unpublished), Majid =-=[36]-=- and Joyal and Street [20]. (See Section 3 for the definition.) In order to avoid confusion with another notion of center, we will write Z1(C) throughout. In the present work, as in [36, 20], we will ... |

24 | Spherical categories
- Barrett, Westbury
- 1999
(Show Context)
Citation Context ...exts like low dimensional topology and subfactor theory. We assume C to be linear over a ground field which is algebraically closed. Furthermore, C is semisimple with simple tensor unit and spherical =-=[6]-=-. (A semisimple category is spherical iff it is pivotal [6] (=sovereign) and every simple object has the same dimension as its dual, cf. [38, Lemma 2.8].) See [6] or [38, Section 2] for the precise de... |

23 |
Drinfel’d: Quantum Groups
- G
- 1986
(Show Context)
Citation Context ...and we briefly comment on this in order to put our results into their context. We recall that the quantum double of a Hopf algebra was introduced, among many other things, in Drinfel’d’s seminal work =-=[11]-=-. In the following discussion all Hopf algebras are finite dimensional over some field F. The quantum double D(H) of a Hopf algebra H is a certain Hopf algebra which contains H and the dual ˆ H as Hop... |

22 |
Turaev, Quantum invariants of knots and 3–manifolds, de Gruyter
- G
- 1994
(Show Context)
Citation Context ...inear tensor category with End(1) ∼ = F. We assume that C is semisimple with finitely many simple objects and dim C ̸= 0. Then also the center Z1(C) has all these properties and is a modular category =-=[53]-=-. Furthermore, the dimension and the Gauss sums are given by dim Z1(C) = (dim C) 2 , ∆+(Z1(C)) = ∆−(Z1(C)) = dim C. 2Defining the center Z2(C) of a braided tensor category C to be the full subcategor... |

22 |
Woronowicz: Compact matrix pseudogroups
- L
- 1987
(Show Context)
Citation Context ...le. But S is nothing but the modular matrix (A.7) as we have i Sij = di µ(Pi S+(Pj)) = di (µ ⊗ µ)(R21R12 (Pi ⊗ Pj)) = 1 (Tr dj q i ⊗ Trq j )(R21R12) We have used µ(Pi) = 1/di and µ(xPi) = 1/di Tri(x) =-=[55]-=-. Note that the proof is conceptually quite similar to our proof of modularity for general semisimple spherical categories. In view of Lemma A.1 the following is now immediate: Corollary A.12 Let H be... |

21 |
An analogue of Longo’s canonical endomorphism for bimodule theory and its application to asymptotic inclusions
- Masuda
- 1997
(Show Context)
Citation Context ...actor M and a – in our language – finite dimensional full monoidal subcategory C of End(M) and conjectured that it is related to Ocneanu’s construction. This conjecture was made precise and proven in =-=[37]-=-. The author’s involvement in the present story began when in 1998 he received a copy of a short preprint [18] by M. Izumi. In the meantime a full account of Izumi’s results has appeared in [19]. In [... |

18 | On finite-dimensional semisimple and cosemisimple Hopf algebras in positive characteristic
- Etingof, Gelaki
- 1998
(Show Context)
Citation Context ...If the characteristic of k is zero then H is semisimple iff it is cosemisimple, and the second condition in (ii) is vacuous. ✷ Proof. For the equivalences (i)⇔(iii)⇔(iv) see [47] and for (i)⇔(ii) see =-=[14]-=-. In order for the category Rep D(H) to be modular it must be semisimple, which by the lemma reduces us to the case where H satisfies (i) and (ii). Lemma A.3 Let H satisfy the (equivalent) conditions ... |

17 | Genealogy of nonperturbative quantum-invariants of 3–manifolds: The surgical family, from: “Geometry and physics
- Kerler
- 1997
(Show Context)
Citation Context ...: z ↦→ 〈z ⊗ id,I〉, I = R21R is injective, thus invertible. Furthermore, it was shown that every quantum double D(H) is factorizable. The notion of factorizability plays an important role in the works =-=[34, 26]-=- where an action of SL(2, Z) on ribbon Hopf algebras is defined and studied. Definition A.6 ([34]) For a quasitriangular Hopf algebra the selfdual Fourier transforms S+, S− are defined by the linear e... |

17 |
An invariant coupling between 3-manifolds and subfactors, with connections to topological and conformal quantum field theory. Unpublished manuscript. Ca
- Ocneanu
- 1991
(Show Context)
Citation Context ...ed in [53]. Later it was understood that in fact no braiding is necessary for the construction of a triangulation invariant, cf. [5, 16], provided C has two-sided duals. (This had been anticipated in =-=[44]-=-, which was never published.) We denote the corresponding invariant by Tr(M, C). Gelfand and Kazhdan formulated a conjecture [16, Conjecture 1] pointing towards a link between the two invariants being... |

12 | A necessary and sufficient condition for a finite-dimensional Drinfel’d double to be a ribbon Hopf algebra - Kauffman, Radford - 1993 |

12 |
Framed tangles and a theorem of Deligne on braided deformations of Tannakian categories
- Yetter
- 1992
(Show Context)
Citation Context ...dimensional and braided then the Gauss sums of C are given by ∆±(C) = ∑ ω(Xi) ±1 d(Xi) 2 , i∈Γ i∈Γ where θ(X) = ω(X)idX is the twist of the simple object X which is defined by the spherical structure =-=[56]-=-. We can now state our Main Theorem: Theorem 1.2 Let F be an algebraically closed field and C a spherical F-linear tensor category with End(1) ∼ = F. We assume that C is semisimple with finitely many ... |

11 |
Some properties of finite dimensional semisimple Hopf algebras
- Etingof, Gelaki
- 1998
(Show Context)
Citation Context ...simple then D(H) is semisimple [47]. It then turns out to be also modular and the category D(H)-mod of finite dimensional left D(H)-modules is modular in the sense of Turaev [53]. (This was proved in =-=[13]-=- 3for algebras over algebraically closed fields of characteristic zero, but the latter condition can be dropped. In the appendix we give a general proof.) Furthermore, one clearly has dim Z1(H-mod) =... |

10 |
Modular categories of types
- Beliakova, Blanchet
(Show Context)
Citation Context ...em (i.e. a premodular category [8]) is modular iff the center Z2(C) is trivial, in the sense that all objects of Z2(C) are multiples of the tensor unit. (This was done in [48] for ∗-categories and in =-=[7]-=- for spherical categories with dim C ̸= 0, see also Corollary 7.11 below.) Thus Z2(Z1(C)) is trivial for all C as in the Main Theorem, which is the promised analogue of the 0-categorical observation Z... |

10 | Galois extensions of braided tensor categories and braided crossed G-categories
- Müger
- 2004
(Show Context)
Citation Context ...assumption. Thus the proposition applies. � Remark 7.9 If we knew that K ∨ (C ∩ K ′ ) = C, we could conclude that C is equivalent, as a braided tensor category, to the direct product K⊠(C ∩ K ′ ). In =-=[40]-=- we will prove that this is indeed the case if C is a modular ∗-category. Thus whenever a modular category C contains a modular category K as a full tensor subcategory then C ⊗ ≃br K ⊠ L, where L = C ... |

10 |
The quantum double and related constructions
- Street
- 1998
(Show Context)
Citation Context ... algebras. Here one remark on the notation is in order. In [23] Kassel and Turaev introduced a modified version of the construction of the center Z1(C) and called it the quantum double D(C), see also =-=[52]-=-. Their category is the categorical version of a construction of Reshetikhin (which adjoins a certain square root θ to a quasitriangular Hopf algebra H in order to turn it into a ribbon algebra H(θ)) ... |

9 |
Topological quantum field theories from subfactors
- Kodiyalam, Sunder
- 2001
(Show Context)
Citation Context ...xplained in more detail in [15, Theorem 3.1]. � Remark 8.4 The above argument is only a sketch because the triangulation TQFT in 2+1 dimensions considered in [45, 15] is derived from a subfactor, see =-=[27]-=- for a detailed exposition. Here as in [38, Section 7] we use the fact that the latter is equivalent to the invariant defined in [5, 16]. This is more or less clear, but certainly deserves being made ... |

8 |
Double construction for monoidal categories
- Kassel, Turaev
- 1995
(Show Context)
Citation Context ...eorem can be considered as an extension of the above results to tensor categories which are not necessarily representation categories of Hopf algebras. Here one remark on the notation is in order. In =-=[23]-=- Kassel and Turaev introduced a modified version of the construction of the center Z1(C) and called it the quantum double D(C), see also [52]. Their category is the categorical version of a constructi... |

8 |
Weyl Algebras, Fourier Transformations and Integrals
- Nill
- 1994
(Show Context)
Citation Context ... D(H) ∗ ̸≃ D(H). (For a finite abelian group G we in fact have D(G) ≃ D(G) ∗ ≃ CG ⊗ C(G).) For any finite dimensional Hopf algebra one can use the integrals to define ‘Fourier transforms’ H → ˆ H. In =-=[42]-=- Fourier transforms Fσ,σ ′,σ,σ′ = ±, defined as linear maps H → ˆ H which intertwine certain actions of H on H and ˆ H by multiplication and translation, respectively, were studied systematically and ... |

6 |
Kazhdan: Invariants of three-dimensional manifolds
- Gelfand, D
- 1996
(Show Context)
Citation Context ...ta equivalent. This is an equivalence relation which is considerably weaker than the usual equivalence, yet it implies that A and B have the same dimension and define the same triangulation invariant =-=[5, 16]-=- for 3-manifolds. See [38] for the details. 2. Unfortunately, the above statement of the theorem will not be sufficient for our purposes since beginning in Subsection 4.2 we will make use of the concr... |

4 |
Coste: Invariants of 3-manifolds from finite groups
- Altschuler, A
- 1991
(Show Context)
Citation Context ...Doubles of Finite Dimensional Hopf Algebras The core of this paper was the proof that the quantum doubles of certain tensor categories are modular. That Rep D(H) is modular has been proven for H = CG =-=[2]-=-, where G is a finite group, and for semisimple H over an algebraically closed field k of characteristic zero [13]. A proof which covers also weak Hopf algebras (or finite quantum groupoids) is given ... |

4 | Rivano: Catégories Tannakiennes - Saaveda - 1972 |

3 | Private communication - Bruguières |

3 |
Drinfel’d: On almost cocommutative Hopf algebras
- G
- 1990
(Show Context)
Citation Context ...ep D(H) is now an obvious consequence of [6] where it was shown under the weaker assumption that S 2 is inner. � We briefly recall some results on quasitriangular Hopf algebras. As shown be Drinfel’d =-=[12]-=-, the antipode of a finite dimensional quasitriangular Hopf algebra H is inner, i.e. there is an invertible u ∈ H such that S 2 (A) = uAu −1 . One has the explicit formulae u = m ◦ (S ⊗ id)(R21), u −1... |

3 |
notes by Y. Kawahigashi): Quantum symmetry, differential geometry of finite graphs and classification of subfactors. Lectures given at Tokyo Univ
- Ocneanu
- 1990
(Show Context)
Citation Context ...ides no clues on how to prove Theorem 1.2. This is where subfactor theory enters the present story. Starting from an inclusion N ⊂ M of hyperfinite type II1 factors of finite index and depth, Ocneanu =-=[43]-=- defined an ‘asymptotic subfactor’ B ⊂ A: B = M ∨ (M∞ ∩ M ′ ) ⊂ M∞ = A. (Here N ⊂ M ⊂ M1 ⊂ M2 ⊂ ... is the Jones tower associated with N ⊂ M and M∞ = ∨iMi.) In [45] he argued that a certain monoidal c... |

2 |
Crans: Generalized centers of braided and sylleptic monoidal 2-categories
- E
- 1998
(Show Context)
Citation Context ...logues to the above result in d = 1 and the trivial case d = 0 mentioned in the Introduction. (See [4] for a review of the theory of n-categories.) Thus, considering the center constructions in d = 2 =-=[3, 10]-=-, can one show that Z (2) 3 (Z(2) 2 (C)) is trivial? Here C, Z(2) 2 (C), Z(2) 3 (Z(2) 2 (C)) are (semisimple spherical) braided, sylleptic and symmetric 2-categories, respectively. ✷ 5.3 Computation o... |

2 |
Lyubashenko: Tangles and Hopf algebras in braided categories
- V
- 1995
(Show Context)
Citation Context ...on. ✷ Remark 5.15 There is little doubt that a more conceptual understanding of the above proof (and of the subsequent subsection) can be gained by looking at them in the light of Lyubashenko’s works =-=[32, 33]-=-. The latter also raise the question whether there is a generalization to non-semisimple Noetherian categories. We hope to pursue this elsewhere. ✷ 38Remark 5.16 It is natural to ask whether there ar... |

2 |
Lyubashenko: Modular transformations for tensor categories
- V
- 1995
(Show Context)
Citation Context ...egory must be non-zero.) In the non-semisimple case one might hope to prove that the center of a spherical noetherian category satisfies the non-degeneracy condition on the the braiding introduced in =-=[33]-=-. But the methods of this paper will most likely not apply. The results of the present work can be considered as generalizations of known results concerning Hopf algebras and we briefly comment on thi... |

1 |
Bruguières: Sliding property in sovereign categories
- Altschuler, A
(Show Context)
Citation Context ... C) −1 removed is identically zero on End Z1(C)(X), thus we cannot use it to obtain a conditional expectation. 14✷ 3. The proof uses a special instance of the ‘handle sliding’ which is formalized in =-=[1]-=-. Lemma 3.12 For every X ∈ C we have trX ◦ EX,X = trX, where trX is the trace on EndC(X) provided by the spherical structure. Proof. Let t ∈ HomC(X,X). Using the fact that the spherical structure of Z... |

1 |
The structure of Longo-Rehren inclusions
- Izumi
- 1998
(Show Context)
Citation Context ...t it is related to Ocneanu’s construction. This conjecture was made precise and proven in [37]. The author’s involvement in the present story began when in 1998 he received a copy of a short preprint =-=[18]-=- by M. Izumi. In the meantime a full account of Izumi’s results has appeared in [19]. In [18, 19] Izumi gives an in-depth analysis of the LR-subfactor, in particular its B − B sectors. Seeing [18] the... |

1 |
notes by Y. Kawahigashi): Chirality for operator algebras
- Ocneanu
- 1994
(Show Context)
Citation Context ... of finite index and depth, Ocneanu [43] defined an ‘asymptotic subfactor’ B ⊂ A: B = M ∨ (M∞ ∩ M ′ ) ⊂ M∞ = A. (Here N ⊂ M ⊂ M1 ⊂ M2 ⊂ ... is the Jones tower associated with N ⊂ M and M∞ = ∨iMi.) In =-=[45]-=- he argued that a certain monoidal category associated with B ⊂ A is braided, concluding that the asymptotic subfactor is an ‘analogue’ of Drinfel’d’s quantum double of a Hopf algebra. In fact, Ocnean... |