2002
DOI: 10.1006/aima.2002.2072
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On a q-Analogue of the McKay Correspondence and the ADE Classification of sl̂2 Conformal Field Theories

Abstract: The goal of this paper is to give a category theory based definition and classification of ''finite subgroups in U q ðsl 2 Þ'' where q ¼ e pi=l is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of U q ðsl 2 Þ; 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 b sl sl 2 at level k ¼ l À 2: We show th… Show more

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Cited by 267 publications
(467 citation statements)
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“…The non commutativity of Oc(D 4 ) does not show up in the fundamental twisted partition functions, indeed, although 2ǫ = ǫ2 in this algebra [11] (actually ǫ2 = 2 ′ ǫ), the toric matrices associated with these two points are the same. 25 Classical symmetries of the D4 diagram are described by the non commutative group algebra of the permutation group S3, this non commutativity also shows up at the quantum level in the structure of Oc(D4). We expect Z 2 to act as usual on A 4 but trivially on Oc(D 4 ); identification is a priori Z (x 1 ,0);(x 2 ,0) = Z (3−x 1 ,0);(x 2 ,0) , i.e., Z 0;x = Z 3;x and Z 1;x = Z 2;x ; this can be checked explicitly.…”
Section: Potts Modelmentioning
confidence: 99%
See 1 more Smart Citation
“…The non commutativity of Oc(D 4 ) does not show up in the fundamental twisted partition functions, indeed, although 2ǫ = ǫ2 in this algebra [11] (actually ǫ2 = 2 ′ ǫ), the toric matrices associated with these two points are the same. 25 Classical symmetries of the D4 diagram are described by the non commutative group algebra of the permutation group S3, this non commutativity also shows up at the quantum level in the structure of Oc(D4). We expect Z 2 to act as usual on A 4 but trivially on Oc(D 4 ); identification is a priori Z (x 1 ,0);(x 2 ,0) = Z (3−x 1 ,0);(x 2 ,0) , i.e., Z 0;x = Z 3;x and Z 1;x = Z 2;x ; this can be checked explicitly.…”
Section: Potts Modelmentioning
confidence: 99%
“…All these diagrams (with or without self -fusion) can also be labelled by an integer k, called the level of the diagram and defined by k = κ − 2. A description of the ADE diagrams in terms of representations of quantum subgroups (a quantum analogue of the McKay correspondance) was discussed by [25] in the framework of modular categories.…”
mentioning
confidence: 99%
“…When C is a fusion category coming from quantum su 2 at a root of unity, then the quantum subgroups are given by the ADE Dynkin diagrams. (See [Ocn88,Ocn99,BEK00] for this result in subfactor language, and [KO02,Ost03a,EO04] for the translation of these results into the language of fusion categories and module categories.) Ocneanu has announced the classification of quantum subgroups of the fusion categories coming from quantum su 3 and su 4 [Ocn02] (see [EP09a,EP09b] for details in the su 3 case).…”
Section: Introductionmentioning
confidence: 99%
“…We shall not use Frobenius algebras in our article but we should nevertheless mention that such a point of view was developped in [12], [25] and that [21] proved that E k is monoidal (existence of self-fusion, so that it is a "quantum subgroup", not only a "quantum module") if and only if F is commutative. Terminological warning: in the older paper [21] this commutativity property was part of the definition and such algebras were called rigid, both requirement and terminology were later abandoned.…”
Section: General Presentationmentioning
confidence: 99%
“…There are 120 vectors K n of norm 1 but there are many repetitions in this family: only 16 of them are unequal (and independent), at positions 1,2,4,11,12,14,20,21,22,29,40,48,57,85,141,197. These independent vectors already give us the first 16 columns of the essential matrix E 0 .…”
Section: Non Isotropy Irreducible Conformal Embeddingsmentioning
confidence: 99%