Quantum Galois Correspondence for Subfactors

Kawahigashi, Yasuyuki

doi:10.1006/jfan.1999.3453

Cited by 8 publications

(5 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since N X N and M X M are equivalent systems of endomorphisms by definition, α-induction produces an example of a subequivalent paragroup. That is, for λ ∈ N X N , the subfactors α ± λ (M) ⊂ M are subequivalent to λ(N) ⊂ N. Various examples in [27] arise from this construction. Indeed, the most fundamental example in [27] comes from the Goodman-de la Harpe-Jones subfactor [17, Section 4.5] with index 3 + √ 3.…”

Section: Discussionmentioning

confidence: 99%

“…That is, for λ ∈ N X N , the subfactors α ± λ (M) ⊂ M are subequivalent to λ(N) ⊂ N. Various examples in [27] arise from this construction. Indeed, the most fundamental example in [27] comes from the Goodman-de la Harpe-Jones subfactor [17, Section 4.5] with index 3 + √ 3. In our current setting, this example comes from the conformal inclusion SU(2) 10 ⊂ SO(5) 1 and shows that the two paragroups with principal graph E 6 are subequivalent to the paragroup with principal graph A 11 .…”

Section: Discussionmentioning

confidence: 99%

“…A notion of subequivalent paragroups was introduced in [27]. Since N X N and M X M are equivalent systems of endomorphisms by definition, α-induction produces an example of a subequivalent paragroup.…”

Section: The Left Action On M-n Sectorsmentioning

confidence: 99%

See 2 more Smart Citations

On α-Induction, Chiral Generators and¶Modular Invariants for Subfactors

Böckenhauer¹,

Evans²,

Kawahigashi³

1999

Communications in Mathematical Physics

Self Cite

142

458

View full text Add to dashboard Cite

We consider a type III subfactor N ⊂ M of finite index with a finite system of braided N -N morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply α-induction and, developing further some ideas of Ocneanu, we define chiral generators for the double triangle algebra. Using a new concept of intertwining braiding fusion relations, we show that the chiral generators can be naturally identified with the α-induced sectors. A matrix Z is defined and shown to commute with the S-and Tmatrices arising from the braiding. If the braiding is non-degenerate, then Z is a "modular invariant mass matrix" in the usual sense of conformal field theory. We show that in that case the fusion rule algebra of the dual system of M -M morphisms is generated by the images of both kinds of α-induction, and that the structural information about its irreducible representations is encoded in the mass matrix Z. Our analysis sheds further light on the connection between (the classifications of) modular invariants and subfactors, and we will construct and analyze modular invariants from SU (n) k loop group subfactors in a forthcoming publication, including the treatment of all SU (2) k modular invariants.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

On α-Induction, Chiral Generators and¶Modular Invariants for Subfactors

Böckenhauer¹,

Evans²,

Kawahigashi³

1999

Communications in Mathematical Physics

Self Cite

142

458

View full text Add to dashboard Cite

show abstract

“…Moreover, it is not clear at the moment how to incorporate the type II modular invariants in our framework. Another challanging question, suggested by the treatment of SU (2), concerns a better understanding of the relation between the appearance of modular invariants of SU (n) WZW models and the existence of sub-(equivalent)-paragroups of the paragroups arising from the relevant A-type subfactors ( [29,39]). Of course it will also be interesting to construct the associated fusion graphs also for modular invariants of other Lie groups, e.g.…”

Section: Discussionmentioning

confidence: 99%

Modular Invariants, Graphs and α-Induction for Nets of Subfactors. II

Böckenhauer

Evans

1999

Communications in Mathematical Physics

107

440

View full text Add to dashboard Cite

We apply the theory of α-induction of sectors which we elaborated in our previous paper to several nets of subfactors arising from conformal field theory. The main application are conformal embeddings and orbifold inclusions of SU (n) WZW models. For the latter, we construct the extended net of factors by hand. Developing further some ideas of F. Xu, our treatment leads canonically to certain fusion graphs, and in all our examples we rediscover the graphs Di Francesco, Petkova and Zuber associated empirically to the corresponding SU (n) modular invariants. We establish a connection between exponents of these graphs and the appearance of characters in the block-diagonal modular invariants, provided that the extended modular S-matrices diagonalize the endomorphism fusion rules of the extended theories. This is proven for many cases, and our results cover all the block-diagonal SU (2) modular invariants, thus provide some explanation of the A-D-E classification.

show abstract

“…C. Dong and G. Mason initiated a systematic search for a vertex operator algebra with a finite automorphism group, which is referred to as the operator content of orbifold models by physicists or as quantum Galois theory for finite groups [6], [9]. There are other Galois correspondences which are relevant to quantum Galois theory for finite groups, especially in the context of subfactors [11], [12].…”

Section: Introductionmentioning

confidence: 99%