2-supertransitive subfactors at index<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mn>3</mml:mn><mml:mo>+</mml:mo><mml:msqrt><mml:mn>5</mml:mn></mml:msqrt></mml:math>

Morrison, Scott; Penneys, David

doi:10.1016/j.jfa.2015.06.023

Cited by 11 publications

(9 citation statements)

References 38 publications

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“…As ρ ≇ µ we have that T T 3 is a rank preserving quotient. Further, one has from [27,Lemma 3.4] that the planar algebra for T T 3 is generated by a single 6-box T . From the relations for T given in the same paper, we can see that the planar algebra for T T 3 has a *-automorphism mapping T ↦ −T .…”

Section: Free Products and Quotientsmentioning

confidence: 99%

A Complete Classification of Unitary Fusion Categories Tensor Generated by an Object of Dimension

Edie-Michell

2020

International Mathematics Research Notices

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In this paper we give a complete classification of unitary fusion categories ⊗generated by an object of dimension 1+ √ 5 2 . We show that all such categories arise as certain wreath products of either the Fibonacci category, or of the dual even part of the 2D2 subfactor. As a by-product of proving our main classification result we produce a classification of finite unitarizable quotients of Fib * N satisfying a certain symmetry condition.

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Section: Free Products and Quotientsmentioning

confidence: 99%

A Complete Classification of Unitary Fusion Categories Tensor Generated by an Object of Dimension

Edie-Michell

2020

International Mathematics Research Notices

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“…Its even part is the Z/2Z de-equivariantization of the principal even part of the 3 Z/4Z subfactor Figure 5. Principal graph of the 2D2 subfactor described in the previous subsection; it was also constructed using planar algebra methods in [MP15a].…”

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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“…Corollary 9.3. There exists a subfactor of index 3 + √ 5 with one of the principal graph as in Figure 1 (see [37]). There exists a unique solution of Eq.…”

Section: Examplesmentioning

confidence: 99%

“…The classification list [1] of small index subfactors shows that there are relatively few finite depth subfactors. However, 3 + √ 5 is an exceptionally rich index value, and there are exactly 4 finite depth subfactors, up to dual, without non-trivial intermediate subfactors (see [36], [37]). Our method gives uniform construction of them.…”

Section: Introductionmentioning

confidence: 99%

The classification of $3^n$ subfactors and related fusion categories

Izumi

2018

Quantum Topol.

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We investigate a (potentially infinite) series of subfactors, called 3 n subfactors, including A 4 , A 7 , and the Haagerup subfactor as the first three members corresponding to n = 1, 2, 3. Generalizing our previous work for odd n, we further develop a Cuntz algebra method to construct 3 n subfactors and show that the classification of the 3 n subfactors and related fusion categories is reduced to explicit polynomial equations under a mild assumption, which automatically holds for odd n. In particular, our method with n = 4 gives a uniform construction of 4 finite depth subfactors, up to dual, without intermediate subfactors of index 3 + √ 5. It also provides a key step for a new construction of the Asaeda-Haagerup subfactor due to Grossman, Snyder, and the author.

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2-supertransitive subfactors at index3+5

Cited by 11 publications

References 38 publications

A Complete Classification of Unitary Fusion Categories Tensor Generated by an Object of Dimension

A Complete Classification of Unitary Fusion Categories Tensor Generated by an Object of Dimension

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

The classification of $3^n$ subfactors and related fusion categories

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