## Orbifold subfactors from Hecke algebras (1994)

Venue: | Comm. Math. Phys |

Citations: | 39 - 23 self |

### BibTeX

@ARTICLE{Evans94orbifoldsubfactors,

author = {David E. Evans and Yasuyuki Kawahigashi},

title = {Orbifold subfactors from Hecke algebras},

journal = {Comm. Math. Phys},

year = {1994},

volume = {165},

pages = {445--484}

}

### OpenURL

### Abstract

A. Ocneanu has observed that a mysterious orbifold phenomenon occurs in the system of the M∞-M ∞ bimodules of the asymptotic inclusion, a subfactor analogue of the quantum double, of the Jones subfactor of type A2n+1. We show that this is a general phenomenon and identify some of his orbifolds with the ones in our sense as subfactors given as simultaneous fixed point algebras by working on the Hecke algebra subfactors of type A of Wenzl. That is, we work on their asymptotic inclusions and show that the M∞-M ∞ bimodules are described by certain orbifolds (with ghosts) for SU(3)3k. We actually compute several examples of the (dual) principal graphs of the asymptotic inclusions. As a corollary of the identification of Ocneanu’s orbifolds with ours, we show that a non-degenerate braiding exists on the even vertices of D2n, n>2. 1

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Citation Context ...f the asymptotic inclusions of the Jones subfactors if the above nondegeneracy condition fails. In this paper, we will generalize his construction to the case of the Hecke algebra subfactors of Wenzl =-=[48]-=- and show that this is a general phenomenon in the following sense. The asymptotic inclusion produces a non-degenerate system of bimodules in the sense of Ocneanu [36]. From the viewpoint of [36], we ... |

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Citation Context ...system of the Q∞-Q∞ bimodules. We thus get a braiding naturally. The non-degeneracy is also easy to see, because if have degeneracy, then the degenerate subsystem would give a finite abelian group by =-=[5]-=-, which is impossible by n>2. Q.E.D. Corollary 7.2 The dual principal graph of the asymptotic inclusion of the hyperfinite II1 subfactor N ⊂ M with principal graph D2n is the fusion graph of the syste... |

104 |
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- 1988
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Citation Context ...s with ours, we show that a non-degenerate braiding exists on the even vertices of D2n, n>2. 1 Introduction In the theory of subfactors initiated by V. F. R. Jones in [17], Ocneanu’s paragroup theory =-=[30]-=- is fundamental in descriptions of the combinatorial structures arising from subfactors. Ocneanu’s construction of the asymptotic inclusions, introduced in [30], has recently caught much attention as ... |

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Citation Context ...2, Z). (This is a direct analogue of the Verlinde formula [46]. Actually, the formula in Theorem 5.1 in [10] is slightly incorrect because normalizing coefficients are missing there. Theorem 12.29 in =-=[11]-=- is correct.) By Lemma 3.7, we can apply this identity to the Ocneanu elements. Then we have pa,b = pa,b ∗ p0,0 = pa,b ∗ pσa,σ n−1 b = pσa,σ n−1 b. Q.E.D. Lemma 3.9 Let a, b, a ′ ,b ′ be primary field... |

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Citation Context ...f the asymptotic inclusions, introduced in [30], has recently caught much attention as a subfactor analogue of the quantum double construction of Drinfel ′ d in [4]. (See [7], [10], [16], [21], [22], =-=[26]-=-, [28], [41] on asymptotic inclusions.) As noted by Ocneanu, if we start with a subfactor N ⊂ M = N ⋊ G, where N is a hyperfinite II1 factor with a free action of a finite group G on N, then the resul... |

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Citation Context ...kin diagrams, only the even D2n appear as paragroups; the odd D2n+1 cannot appear as was announced in [O1] and proved in [Ka]. (Izumi gave a different proof of impossibility of D2n+1 independently in =-=[I]-=- based on Longo’s sector theory [Ln1, Ln2], and its bimodule version was given by Evans-Gould in [EG2] and by Sunder-Vijayarajan in [SV] independently. This approach was also claimed by Ocneanu withou... |

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Citation Context ...is follows from the above Lemma, because we have N c σ(c) ab = Naσ(b) by [47]. Q.E.D. We now need some lemmas for the fusion rule of the WZW-model SU(3)3k, which has been obtained by Goodman–Wenzl in =-=[14]-=- as a quantum version of the classical Littlewood–Richardson rule. Each primary field is represented by a Young diagram and we denote a primary field by the corresponding Young diagram. Lemma 5.4 We h... |

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Citation Context ...in [36], [11, Sections 12.7]. The dual principal graphs of the asymptotic inclusions are hard to compute, in general, while their principal graphs are easy to compute, as in [10], [11, Section 12.6], =-=[31]-=-, [32], [33], as long as we know the fusion rule of the M-M bimodules of the original subfactor N ⊂ M. From the above viewpoint of the quantum double, it is the dual principal graph, or the system of ... |

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Citation Context ... the vertices with symmetries on pairs of the original labels. In the SU(2)k case, Ocneanu has noticed that this situation is similar to the orbifold construction for subfactors studied by us in [8], =-=[20]-=-. (See also [15], [51].) However, the dual principal graphs we have studied in Sections 4, 5 are not orbifold graphs in the sense of [8], [20], [51], because we have merging/splitting of the vertices ... |

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Coxeter graphs and towers of algebras, MSRI publications 14
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rules of Wess-Zumino-Witten Models
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17 |
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- 1991
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- 1990
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