Constructing spoke subfactors using the jellyfish algorithm

Morrison, Scott; Penneys, David

doi:10.1090/s0002-9947-2014-06109-6

Cited by 21 publications

(44 citation statements)

References 17 publications

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“…The point of working with sets of minimal generators is the following theorem, first observed in [MP12a].…”

Section: Theorem 22 ([Jp11 Mw])mentioning

confidence: 99%

“…Since we do not assume our generators in B are orthonormal, our formulas will differ slightly in appearance than those of [Jon12] and [MP12a].…”

Section: Theorem 22 ([Jp11 Mw])mentioning

confidence: 99%

“…By now, these methods have been used to construct a large handful of examples, some new and some well known, including the E 6 and E 8 subfactors [Jon01], group-subgroup subfactors [Gup08], the Haagerup subfactor [Pet10], the extended Haagerup subfactor [BPMS12], the Izumi-Xu 2221 subfactor [Han10], and certain spoke subfactors, e.g., 4442 [MP12a]. These techniques have also been used to prove uniqueness results [BPMS12,Han10] and obstructions to possible principal graphs [Pet10,Mor13].…”

Section: Introductionmentioning

confidence: 99%

“…If both graphs are spokes, one can use one-strand relations, which are easier to compute. Recent work of Morrison-Penneys [MP12a] found an algorithm to compute these one-strand relations, provided one has the generators in the graph planar algebra.…”

Section: Introductionmentioning

confidence: 99%

“…This article is the natural followup to [BP13,MP12a]. The main result of this article is an algorithm to find two-strand jellyfish relations for a subfactor planar algebra for which one of the principal graphs is a spoke graph, provided we are given the generators in a graph planar algebra.…”

Section: Introductionmentioning

confidence: 99%

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Calculating two-strand jellyfish relations

Penneys

Peters

2015

Pacific J. Math.

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We construct subfactors where one of the principal graphs is a spoke graph using an algorithm which computes two-strand jellyfish relations. One of the subfactors we construct is a 3 Z/4 subfactor known to Izumi, which has not previously appeared in the literature. To do so, we provide a systematic treatment of the space of second annular consequences, which is analogous to Jones' treatment of the space of first annular consequences in his quadratic tangles article.This article is the natural followup to two recent articles on spoke subfactor planar algebras and the jellyfish algorithm. Work of Bigelow-Penneys explains the connection between spoke subfactor planar algebras and the jellyfish algorithm, and work of Morrison-Penneys automates the construction of subfactors where both principal graphs are spoke graphs using one-strand jellyfish.

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“…The point of working with sets of minimal generators is the following theorem, first observed in [MP12a].…”

Section: Theorem 22 ([Jp11 Mw])mentioning

confidence: 99%

“…Since we do not assume our generators in B are orthonormal, our formulas will differ slightly in appearance than those of [Jon12] and [MP12a].…”

Section: Theorem 22 ([Jp11 Mw])mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Calculating two-strand jellyfish relations

Penneys

Peters

2015

Pacific J. Math.

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Chirality and principal graph obstructions

Penneys

2015

Advances in Mathematics

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Determining which bipartite graphs can be principal graphs of subfactors is an important and difficult question in subfactor theory. Using only planar algebra techniques, we prove a triple point obstruction which generalizes all known initial triple point obstructions to possible principal graphs. We also prove a similar quadruple point obstruction with the same technique. Using our obstructions, we eliminate some infinite families of possible principal graphs with initial triple and quadruple points which were a major hurdle in extending subfactor classification results above index 5.

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Jones-Wassermann subfactors for modular tensor categories

Liu

2019

Advances in Mathematics

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The representation theory of a conformal net is a unitary modular tensor category. It is captured by the bimodule category of the Jones-Wassermann subfactor. In this paper, we construct multi-interval Jones-Wassermann subfactors for unitary modular tensor categories. We prove that these subfactors are self-dual. It generalizes and categorifies the self-duality of finite abelian groups and we call it modular self-duality. arXiv:1612.08573v3 [math.OA] 8 Jun 2017 ZHENGWEI LIU AND FENG XUThe bimodule category of a subfactor is described by a subfactor planar algebra [Jon98]. The n-box space of the planar algebra of the m-interval Jones-Wassermann subfactor for C is given by the vector space hom C m (1, γ n m ). It turns out to be natural to represent these vectors by a 3D picture. This representation identifies hom C m (1, γ n m ) as a configuration space Conf n,m on a 2D n × m lattice. Therefore the configuration space {Conf n,m } m,n∈N unifies the Jones-Wassermann subfactors for all m ≥ 1. It is a natural candidate for the configuration space of a 2D lattice model that can be used in the reconstruction program.Moreover, we show that planar tangles can act on {Conf n,m } m,n∈N in two different directions independently. In one direction m is fixed. These actions are the usual ones in the planar algebra of the m-interval Jones-Wassermann subfactor. In the other direction n is fixed. These actions relate the Jones-Wassermann subfactor with different intervals which have not been studied before.The bi-directional actions of planar tangles are compatible with the geometric actions on the 2D lattices. We call such family of vector spaces a bi-planar algebra. It is a new subject in subfactor theory and it adds one additional dimension to the theory of planar algebras.This 3D representation also leads to the discovery of a new symmetry between m and n, although the meaning of the actions of planar tangles in the two directions are completely different. It will be interesting to understand these additional symmetries in conformal field theory.When C is the representation category of a finite abelian group G, the configuration space Conf (C ) 2,2 becomes L 2 (G). Moreover, the modular self-duality coincides with the self-duality of G. The proof of the self-duality of G uses the discrete Fourier transform on G. We construct the string Fourier transform (SFT) on the configuration space hom C m (1, γ n m ) to prove the modular self-duality. From this point of view, the modular self-duality and the SFT generalize and categorify the self-duality and the Fourier transform of finite abelian groups.Moreover, the SFT on Conf (C ) 2,2 is the same as the modular S-matrix of C . Therefore we can study the Fourier analysis of the S-matrix on Conf (C ) 2,2 . It fits into the recent progress about the Fourier analysis on subfactors [Liu16, JLW16, LW17, JLW].The modular self-duality has been used in the quon 3D language for quantum information [LWJ], where the vector space hom C 2 (1, γ 2 2 ) is considered as the 1-quon space. A combination ...

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Constructing spoke subfactors using the jellyfish algorithm

Cited by 21 publications

References 17 publications

Calculating two-strand jellyfish relations

Calculating two-strand jellyfish relations

Chirality and principal graph obstructions

Jones-Wassermann subfactors for modular tensor categories

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