Representations of fusion categories and their commutants

Henriques, André; Penneys, David

doi:10.1007/s00029-023-00841-2

Cited by 3 publications

(2 citation statements)

References 36 publications

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“…In [HP23,§5], we introduced a certain extra structure on Bim(R) which we called a positive structure. These positive structures play no role in the present paper, as representations of unitary fusion categories uniquely extend to positive representations by [HP23, Thm.…”

Section: Bicommutant Categoriesmentioning

confidence: 99%

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Planar Algebras in Braided Tensor Categories

Henriques¹,

Penneys²,

Tener³

2023

Memoirs of the AMS

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We generalize Jones’ planar algebras by internalising the notion to a pivotal braided tensor category C \mathcal {C} . To formulate the notion, the planar tangles are now equipped with additional ‘anchor lines’ which connect the inner circles to the outer circle. We call the resulting notion an anchored planar algebra. If we restrict to the case when C \mathcal {C} is the category of vector spaces, then we recover the usual notion of a planar algebra. Building on our previous work on categorified traces, we prove that there is an equivalence of categories between anchored planar algebras in C \mathcal {C} and pivotal module tensor categories over C \mathcal {C} equipped with a chosen self-dual generator. Even in the case of usual planar algebras, the precise formulation of this theorem, as an equivalence of categories, has not appeared in the literature. Using our theorem, we describe many examples of anchored planar algebras.

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Section: Bicommutant Categoriesmentioning

confidence: 99%

“…Let R be a hyperfinite factor of type II 1 , III ∞ , or III 1 . Let C ⊂ Bim(R) be a unitary fusion subcategory, embedded in Bim(R) via a fully faithful unitary tensor functor, and let C ′ be its commutant category (see [HP17,HP23], or Definition 2.5 below). The main result of this paper states that Conjecture 1.1 is true when B = C ′ .…”

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confidence: 99%

Planar Algebras in Braided Tensor Categories

Henriques¹,

Penneys²,

Tener³

2023

Memoirs of the AMS

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Unitary Anchored Planar Algebras

Henriques,

Penneys,

Tener

2024

Commun. Math. Phys.

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In our previous article (http://arxiv.org/abs/1607.06041), we established an equivalence between pointed pivotal module tensor categories and anchored planar algebras. This article introduces the notion of unitarity for both module tensor categories and anchored planar algebras, and establishes the unitary analog of the above equivalence. Our constructions use Baez’s 2-Hilbert spaces (i.e., semisimple $$\textrm{C}^*$$ C ∗ -categories equipped with unitary traces), the unitary Yoneda embedding, and the notion of unitary adjunction for dagger functors between 2-Hilbert spaces.

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A categorical Connes' $χ(M)$

Chen¹,

Jones²,

Penneys³

2021

Preprint

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Popa introduced the tensor category χ(M ) of approximately inner, centrally trivial bimodules of a II1 factor M , generalizing Connes' χ(M ). We extend Popa's notions to define the W * -tensor category End loc (C) of local endofunctors on a W * -category C. We construct a unitary braiding on End loc (C), giving a new construction of a braided tensor category associated to an arbitrary W * -category. For the W * -category of finite modules over a II1 factor, this yields a unitary braiding on Popa's χ(M ), which extends Jones' κ invariant for χ(M ).Given a finite depth inclusion M0 ⊆ M1 of non-Gamma II1 factors, we show that the braided unitary tensor category χ(M∞) is equivalent to the Drinfeld center of the standard invariant, where M∞ is the inductive limit of the associated Jones tower. This implies that for any pair of finite depth non-Gamma subfactors N0 ⊆ N1 and M0 ⊆ M1, if the standard invariants are not Morita equivalent, then the inductive limit factors N∞ and M∞ are not stably isomorphic.

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Representations of fusion categories and their commutants

Cited by 3 publications

References 36 publications

Planar Algebras in Braided Tensor Categories

Planar Algebras in Braided Tensor Categories

Unitary Anchored Planar Algebras

A categorical Connes' $χ(M)$

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