Calculating two-strand jellyfish relations

Penneys, David; Peters, Emily

doi:10.2140/pjm.2015.277.463

Cited by 10 publications

(4 citation statements)

References 24 publications

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“…The 3 Z/2Z×Z/2Z subfactor is self-dual and the 3 Z/4Z subfactor is not. These subfactors have also been constructed using planar algebra methods [MP15b,PP15].…”

Section: 5mentioning

confidence: 99%

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

Grossman¹

2019

Indiana Univ. Math. J.

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We classify fusion categories which are Morita equivalent to even parts of subfactors with index 3 + √ 5, and module categories over these fusion categories. For the fusion category C which is the even part of the self-dual 3 Z/2Z×Z/2Z subfactor, we show that there are 30 simple module categories over C; there are no other fusion categories in the Morita equivalence class; and the order of the Brauer-Picard group is 360. The proof proceeds indirectly by first describing the Brauer-Picard groupoid of a Z/3Z-equivariantization C Z/3Z (which is the even part of the 4442 subfactor). We show that that there are exactly three other fusion categories in the Morita equivalence class of C Z/3Z , which are all Z/3Z-graded extensions of C. Each of these fusion categories admits 20 simple module categories, and their Brauer-Picard group is S 3 . We also show that there are exactly five fusion categories in the Morita equivalence class of the even parts of the 3 Z/4Z subfactor; each admits 7 simple module categories; and the Brauer-Picard group is Z/2Z.

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“…The 3 Z/2Z×Z/2Z subfactor is self-dual and the 3 Z/4Z subfactor is not. These subfactors have also been constructed using planar algebra methods [MP15b,PP15].…”

Section: 5mentioning

confidence: 99%

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

Grossman¹

2019

Indiana Univ. Math. J.

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“…The unique 2D2 subfactor planar algebra from Theorem 1.1 has a planar automorphism by mapping the uncappable rotational eigenvector T at depth 3 to its negative. The fixed point subfactor planar algebra under this automorphism is Izumi's 3 Z/4Z subfactor planar algebra constructed in [Izu,PP13].…”

Section: Recall the Conjecture Of Morrison-peters [Mp12b]mentioning

confidence: 99%

“…whereǍ := −iF(A) (as in [PP13], due to Lemma 4.4). Note e 1 4,− + e 2 4,− = g 1 and e 3 4,− + e 4 4,− = g 2 .…”

Section: The Dual Graphmentioning

confidence: 99%

2-supertransitive subfactors at index3+5

Morrison

Penneys

2015

Journal of Functional Analysis

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“…There are exactly seven finite depth subfactor planar algebras at index 3 C p 5 up to duality [1]. Of these, two are the unique 3 Z 4 and 3 Z 2 Z 2 subfactors; another one is the 2D2 subfactor , which is related to the 3 Z 4 subfactor through a Z 2 -de-equivariantization; and another one is the 4442 subfactor, which is related to the 3 Z 2 Z 2 subfactor through a Z 3 -equivariantization (and another one is related to the 3 Z 4 subfactor by Morita equivalence), see [19,26,27,32].…”

Section: Introductionmentioning

confidence: 99%

Drinfeld centers of fusion categories arising from generalized Haagerup subfactors

Grossman

Izumi

2023

Quantum Topol.

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We consider generalized Haagerup categories such that 1 ˚X admits a Q-system for every non-invertible simple object X . We show that in such a category, the group of order two invertible objects has size at most four. We describe the simple objects of the Drinfeld center and give partial formulas for the modular data. We compute the remaining corner of the modular data for several examples and make conjectures about the general case. We also consider several types of equivariantizations and de-equivariantizations of generalized Haagerup categories and describe their Drinfeld centers.In particular, we compute the modular data for the Drinfeld centers of a number of examples of fusion categories arising in the classification of small-index subfactors: the Asaeda-Haagerup subfactor; the 3 Z 4 and 3 Z 2 Z 2 subfactors; the 2D2 subfactor; and the 4442 subfactor.The results suggest the possibility of several new infinite families of quadratic categories. A description and generalization of the modular data associated to these families in terms of pairs of metric groups is taken up in the accompanying paper [Comm. Math. Phys. 380 (2020), 1091-1150.

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Calculating two-strand jellyfish relations

Cited by 10 publications

References 24 publications

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

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

2-supertransitive subfactors at index3+5

Drinfeld centers of fusion categories arising from generalized Haagerup subfactors

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