Higher Coxeter graphs associated with affine <i>su</i>(3) modular invariants

Hammaoui, D.; Schieber, Gil; Tahri, E. H.

doi:10.1088/0305-4470/38/38/007

Cited by 11 publications

(29 citation statements)

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“…Therefore we fix K and look for "overgroups" G such that the pair (K, G) is conformal. The fact that each such pair gives rise to a quantum subgroup of K results from investigations carried out more recently (in the last ten years) but we should stress than few of them have been worked out explicitly: only the SU (N ) cases with N = 2, 3, 4 are described (their associated graphs and algebras of quantum symmetries are known) in the available literature [23,4,26,24,5,14,15,6,7]. We always assume that G is simple.…”

Section: Quantum Subgroups Of Lie Groups From Conformal Embeddings 4mentioning

confidence: 99%

“…When 6 N = 2, 3, 4 we have only the adjoint embedding into SO (N (2N + 1)), at level N +1, and also an embedding into SO (N (2N +1)) at level N −1. The case N = 2 coincides with SO(5) so that we have conformal embeddings into SU (5), SO(10), SO (14) and E 8 at respective levels 2, 3, 7 and 12. When N = 3, we have also a non standard exceptional E 5 (Sp(3)) coming from an embedding into Sp (7), and when N = 4 one obtains two other quantum subgroups : E 1 (Sp (4)) from an embedding into E 6 , and E 7 (Sp (4)) from an embedding into SO(42).…”

Section: So(2n +1)mentioning

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%

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Conformal embeddings and quantum graphs with self-fusion

Coquereaux

2009

São Paulo J. Math. Sci.

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Section: Quantum Subgroups Of Lie Groups From Conformal Embeddings 4mentioning

confidence: 99%

Section: So(2n +1)mentioning

confidence: 99%

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Conformal embeddings and quantum graphs with self-fusion

Coquereaux

2009

São Paulo J. Math. Sci.

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“…In particular we give in most cases an algebraic realization of Oc(G) that allows one to perform calculations without having to use the graph of quantum symmetries. A detailed study of several cases has already been made available in the litterature [13,47] and details concerning the others will be published elsewhere [25,27,24]. Several graphs are displayed in figures 4 and 5.…”

Section: Realization Of the Ocneanu Quantum Symmetriesmentioning

confidence: 99%

“…In the fifth section we describe the equations that allow one to recover the algebra of quantum symmetries (and sometimes the graph itself) from the data provided by a modular invariant, the leitmotiv of this section being the so-called "modular splitting technique". Although we have used repeatedly this technique to solve several quite involved examples briefly described in section 6, we do not explicitly discuss here our method of resolution but refer to forthcoming articles (or theses) for these -important -details [27,26,24]. In section 6 we summarize what is known, or at least what we know, about the structure of the algebra of quantum symmetries for each member of the SU (3) series.…”

Section: Introductionmentioning

confidence: 99%

Comments about quantum symmetries of SU(3) graphs

Coquereaux¹,

Hammaoui²,

Schieber³

et al. 2006

Journal of Geometry and Physics

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Orders and dimensions for sl(2) or sl(3) module categories and boundary conformal field theories on a torus

Coquereaux

Schieber

2007

Journal of Mathematical Physics

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After giving a short description, in terms of action of categories, of some of the structures associated with sl(2) and sl(3) boundary conformal field theories on a torus, we provide tables of dimensions describing the semisimple and co-semisimple blocks of the corresponding weak bialgebras (quantum groupoids), tables of quantum dimensions and orders, and tables describing induction -restriction. For reasons of size, the sl(3) tables of induction are only given for theories with self-fusion (existence of a monoidal structure).II, et du Sud Toulon-Var, affiliéà la FRUMAM (FR 2291). ‡ Email: schieber@cbpf.br § CBPF -Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro. The stageIn this paper A k is the fusion category of the affine algebra sl(2), or sl (3), at level k, or equivalently, the category of representations with non-zero q-dimension for the quantum groups SL(2) q or SL(3) q at roots of unity (set q = exp(iπ/κ), with κ = k + 2 for sl(2) and κ = k + 3 for sl (3)). This category is additive (existence of ⊕), monoidal (existence of ⊗ : A k × A k → A k , with associativity constraints, unit object, etc.), tensorial (⊗ is a bifunctor), complex-linear, rigid (existence of duals), finite (finitely many irreducible objects), and semisimple, with irreducible unit object. It is also modular (braided, with invertible S-matrix) and ribbon (in particular balanced -or tortile). We refer to the literature [29], [27], [1] for a detailed description of these structures. The Grothendieck ring of this monoidal category comes with a special basis (corresponding to simple objects), it is usually called the fusion ring, or the Verlinde algebra. The corresponding structure constants, encoded by the so -called fusion matrices (N n ) p q , are therefore non -negative integers: NIM-reps in CFT terminology. The rigidity property of the category implies that (N n ) pq = (N n ) qp , where n refers to the dual object i.e., in our case, to the conjugate representation, so that the fusion ring is automatically a based Z + ring in the sense of [39] (maybe it would be better to call it "rigid"). In the case of sl (2), this is a ring with one generator (corresponding to the fundamental representation), and fusion matrices are symmetric, because n = n. In the case of sl (3), it is convenient to use two generators that are conjugate to one another; they correspond to the two fundamental representations. Multiplication by the chosen generator is encoded by a particular fusion matrix N 1 ; it is a finite size matrix of dimension r × r, with r = k + 1 for sl(2) and r = k(k + 1)/2 for sl(3). Since its elements are non negative integers, it can be interpreted as the adjacency matrix of a graph, which is the Cayley graph of multiplication by this generator, that we call the McKay graph of the category. Edges are non oriented in the case of sl(2) (rather, they carry both orientations) and are oriented in the case of sl(3). Irreducible representations are denoted by λ p , with p ≥ 0, in the first case, and λ pq , with p, q ≥ 0 in the next. Notice t...

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Higher Coxeter graphs associated with affine su(3) modular invariants

Cited by 11 publications

References 37 publications

Conformal embeddings and quantum graphs with self-fusion

Conformal embeddings and quantum graphs with self-fusion

Comments about quantum symmetries of SU(3) graphs

Orders and dimensions for sl(2) or sl(3) module categories and boundary conformal field theories on a torus

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