Fusion categories associated with subfactors with index $3 + sqrt{5}$

Grossman, Pinhas

doi:10.1512/iumj.2019.68.7667

Cited by 4 publications

(5 citation statements)

References 23 publications

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“…We now deal with the case of C(𝔰𝔩 5 , 5) 0 Rep(ℤ 5 ) . This case has been examined in the literature previously [32,64]. Rep(ℤ 4 ) .…”

Section: Rep(ℤ𝑚)mentioning

confidence: 99%

Type 𝐼𝐼 quantum subgroups of 𝔰𝔩_{𝔑}. ℑ: Symmetries of local modules

Edie-Michell

2023

Comm. Amer. Math. Soc.

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This paper is the first of a pair that aims to classify a large number of the type I I II quantum subgroups of the categories C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . In this work we classify the braided auto-equivalences of the categories of local modules for all known type I I quantum subgroups of C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . We find that the symmetries are all non-exceptional except for four cases (up to level-rank duality). These exceptional cases are the orbifolds C ( s l 2 , 16 ) Rep ⁡ ( Z 2 ) 0 \mathcal {C}(\mathfrak {sl}_{2}, 16)^0_{\operatorname {Rep}(\mathbb {Z}_{2})} , C ( s l 3 , 9 ) Rep ⁡ ( Z 3 ) 0 \mathcal {C}(\mathfrak {sl}_{3}, 9)^0_{\operatorname {Rep}(\mathbb {Z}_{3})} , C ( s l 4 , 8 ) Rep ⁡ ( Z 4 ) 0 \mathcal {C}(\mathfrak {sl}_{4}, 8)^0_{\operatorname {Rep}(\mathbb {Z}_{4})} , and C ( s l 5 , 5 ) Rep ⁡ ( Z 5 ) 0 \mathcal {C}(\mathfrak {sl}_{5}, 5)^0_{\operatorname {Rep}(\mathbb {Z}_{5})} . We develop several technical tools in this work. We give a skein theoretic description of the orbifold quantum subgroups of C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . Our methods here are general, and the techniques developed will generalise to give skein theory for any orbifold of a braided tensor category. We also give a formulation of orthogonal level-rank duality in the type D D - D D case, which is used to construct one of the exceptionals. We uncover an unexpected connection between quadratic categories and exceptional braided auto-equivalences of the orbifolds. We use this connection to construct two of the four exceptionals. In the sequel to this paper we will use the classified braided auto-equivalences to construct the corresponding type I I II quantum subgroups of the categories C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) . This will essentially finish the type I I II classification for s l n \mathfrak {sl}_n modulo type I I classification. When paired with Gannon’s type I I classification for r ≤ 6 r\leq 6 , our results will complete the type I I II classification for these same ranks. This paper includes an appendix by Terry Gannon, which provides useful results on the dimensions of objects in the categories C ( s l r + 1 , k ) \mathcal {C}(\mathfrak {sl}_{r+1}, k) .

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“…We now deal with the case of C(𝔰𝔩 5 , 5) 0 Rep(ℤ 5 ) . This case has been examined in the literature previously [32,64]. Rep(ℤ 4 ) .…”

Section: Rep(ℤ𝑚)mentioning

confidence: 99%

Type 𝐼𝐼 quantum subgroups of 𝔰𝔩_{𝔑}. ℑ: Symmetries of local modules

Edie-Michell

2023

Comm. Amer. Math. Soc.

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“…We now deal with the case of C(𝔰𝔩 5 , 5) 0 Rep(ℤ 5 ) . This case has been examined in the literature previously [32,64]. Rep(ℤ 4 ) .…”

Section: Rep(ℤ𝑚)mentioning

confidence: 99%

Untitled

2023

Comm. Amer. Math. Soc.

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This paper is the first of a pair that aims to classify a large number of the type 𝐼𝐼 quantum subgroups of the categories C(𝔰𝔩 𝑟+1 , 𝑘). In this work we classify the braided auto-equivalences of the categories of local modules for all known type 𝐼 quantum subgroups of C(𝔰𝔩 𝑟+1 , 𝑘). We find that the symmetries are all non-exceptional except for four cases (up to level-rank duality). These exceptional cases are the orbifolds, and C(𝔰𝔩5, 5) 0 Rep(ℤ 5 ) . We develop several technical tools in this work. We give a skein theoretic description of the orbifold quantum subgroups of C(𝔰𝔩 𝑟+1 , 𝑘). Our methods here are general, and the techniques developed will generalise to give skein theory for any orbifold of a braided tensor category. We also give a formulation of orthogonal level-rank duality in the type 𝐷-𝐷 case, which is used to construct one of the exceptionals. We uncover an unexpected connection between quadratic categories and exceptional braided autoequivalences of the orbifolds. We use this connection to construct two of the four exceptionals.In the sequel to this paper we will use the classified braided auto-equivalences to construct the corresponding type 𝐼𝐼 quantum subgroups of the categories C(𝔰𝔩 𝑟+1 , 𝑘). This will essentially finish the type 𝐼𝐼 classification for 𝔰𝔩𝑛 modulo type 𝐼 classification. When paired with Gannon's type 𝐼 classification for 𝑟 ≤ 6, our results will complete the type 𝐼𝐼 classification for these same ranks.This paper includes an appendix by Terry Gannon, which provides useful results on the dimensions of objects in the categories C(𝔰𝔩 𝑟+1 , 𝑘).

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“…As motivation for studying this type of automorphism, we note that it is shown in [Gro19] that the Brauer-Picard group of the generalized Haagerup subfactor for Z 4 is isomorphic to Z 2 , and is generated by such an outer automorphism. As we will see below, such outer automorphisms also exist for all known examples of generalized Haagerup categories for even groups.…”

Section: Classification Of Extensionsmentioning

confidence: 99%

“…This category is related to a conformal inclusion SU (5) 5 ⊂ Spin(24); see [Xu18;Edi21a]. This category is interesting because its Brauer-Picard group is unusually rich: it was shown in [Gro19] that this group has order 360, and it was identified as S 3 × A 5 in [Edi21a]. The outer automorphism subgroup is A 4 .…”

Section: Introductionmentioning

confidence: 99%

Graded extensions of generalized Haagerup categories

Grossman¹,

Izumi²,

Snyder³

2022

Preprint

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We classify certain Z 2 -graded extensions of generalized Haagerup categories in terms of numerical invariants satisfying polynomial equations. In particular, we construct a number of new examples of fusion categories, including: Z 2 -graded extensions of Z 2n generalized Haagerup categories for all n ≤ 5; Z 2 × Z 2 -graded extensions of the Asaeda-Haagerup categories; and extensions of the Z 2 × Z 2 generalized Haagerup category by its outer automorphism group A 4 . The construction uses endomorphism categories of operator algebras, and in particular, free products of Cuntz algebras with free group C * -algebras.

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Fusion categories associated with subfactors with index $3 + sqrt{5}$

Cited by 4 publications

References 23 publications

Type 𝐼𝐼 quantum subgroups of 𝔰𝔩_{𝔑}. ℑ: Symmetries of local modules

Type 𝐼𝐼 quantum subgroups of 𝔰𝔩_{𝔑}. ℑ: Symmetries of local modules

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Graded extensions of generalized Haagerup categories

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