Frobenius–Schur indicators in Tambara–Yamagami categories

Shimizu, Kenichi

doi:10.1016/j.jalgebra.2011.02.002

Cited by 20 publications

(23 citation statements)

References 26 publications

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“…If C is the category of representations of a semisimple Hopf algebra, then C has a canonical pivotal structure and ν n,1 (V ) agrees with Linchenko and Montgomery's. This generalization has some interesting applications to semisimple Hopf algebras, fusion categories and conformal field theories; see [22,25,29,31].…”

Section: Introductionmentioning

confidence: 99%

The pivotal cover and Frobenius–Schur indicators

Shimizu¹

2015

Journal of Algebra

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Abstract. In this paper, we introduce the notion of the pivotal cover C piv of a left rigid monoidal category C to develop a theoretical foundation for the theory of Frobenius-Schur (FS) indicators in "non-pivotal" settings. For an object V ∈ C piv , the (n, r)-th FS indicator νn,r(V) is defined by generalizing that of an object of a pivotal monoidal category. This notion gives a categorical viewpoint to some recent results on generalizations of FS indicators.Based on our framework, we also study the FS indicators of the "adjoint object" in a finite tensor category, which can be considered as a generalization of the adjoint representation of a Hopf algebra. The indicators of this object closely relate to the space of endomorphisms of the iterated tensor product functor.

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Section: Introductionmentioning

confidence: 99%

The pivotal cover and Frobenius–Schur indicators

Shimizu¹

2015

Journal of Algebra

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“…We note that this expression is essentially the same as the formula for the n th higher Frobenius-Schur indicator of a, which has been derived in the mathematical community by very different methods 42,60 .…”

Section: Discussionmentioning

confidence: 75%

Signatures of broken parity and time-reversal symmetry in generalized string-net models

Lake

2016

Phys. Rev. B

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We study indicators of broken time-reversal and parity symmetries in gapped topological phases of matter. We focus on phases realized by Levin-Wen string-net models, and generalize the stringnet model to describe phases which break parity and time-reversal symmetries. We do this by introducing an extra degree of freedom into the string-net graphical calculus, which takes the form of a branch cut located at each vertex of the underlying string-net lattice. We also work with stringnet graphs defined on arbitrary (non-trivalent) graphs, which reveals otherwise hidden information about certain configurations of anyons in the string-net graph. Most significantly, we show that objects known as higher Frobenius-Schur indicators can provide several efficient ways to detect whether or not a given topological phase breaks parity or time-reversal symmetry.

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“…However, to get a factor planar algebra, we need that m is symmetrically self-dual, i.e., m has Frobenius-Schur indicator 1 [NS07]. The Frobenius-Schur indicators for Tambara-Yamagami categories were completely worked out in [Shi11], where it was shown that ν 2 (a) = δ a 2 ,e and ν 2 (m) = ±, the sign in T Y(A, χ, ±). Hence we must have ± = + to get a factor planar algebra.…”

Section: Examplesmentioning

confidence: 99%

Lifting shadings on symmetrically self-dual subfactor planar algebras

Liu¹,

Morrison²,

Penneys³

2020

Topological Phases of Matter and Quantum Computation

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In this note, we discuss the notion of symmetric self-duality of shaded planar algebras, which allows us to lift shadings on subfactor planar algebras to obtain Z/2Z-graded unitary fusion categories. This nishes the proof that there are unitary fusion categories with fusion graphs 4442 and 3333.Planar algebras have proven to be useful in the construction [Pet10; BMPS12] and classi cation [JMS14; AMP15] of subfactors and fusion categories. In recent articles, we used planar algebras to construct subfactor planar algebras with principal graphs 4442, 3333, and 2221 [MP15], ,, and , and a new subfactor with principal graphs 22221 with interesting dual data [LMP15] (see Example 2.2 below). This 22221 subfactor turns out to be an example of a new parameterized family of unshaded subfactor planar algebras related to quantum subgroups [Liu15]. The 2221 subfactor was originally constructed by Izumi [Izu01], as was the 3333 subfactor [Izu16]. In [CMS11, Appendix], Ostrik constructed a Z/2-graded unitary fusion category with fusion graph 2221. Thus upon constructing 4442 and 3333, we naturally wondered whether we could lift the shading on our subfactor planar algebras to get Z/2-graded fusion categories with these fusion graphs.We showed that these subfactor planar algebras are symmetrically self-dual, i.e., there is a planar algebra isomorphism Φ from P • = (P + , P − ) to its dual P • = (P − , P + ) such that Φ ∓ • Φ ± = 1 ± . We furthermore claimed that we can lift the shading on a symmetrically self-dual subfactor planar algebra to obtain an unshaded factor planar algebra [BHP12]. This would mean the associated tensor category of projections is a Z/2-graded unitary fusion category C whose even graded part is the even part of P • and whose odd graded part is the odd part of P • .In this note, we complete the proof of this claim to complete the construction of these categories.Theorem A. Given a symmetrically self-dual shaded planar algebra ( • , Φ), there is an unshaded planar algebra • such that • is obtained from • by re-shading (as in De nition 1.4 below).Corollary B. There are Z/2-graded unitary fusion categories with fusion graphs 4442 and 3333.arXiv:1709.05023v1 [math.OA]

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Frobenius–Schur indicators in Tambara–Yamagami categories

Abstract: We introduce formulae of Frobenius-Schur indicators of simple objects of Tambara-Yamagami categories. By using techniques of the Fourier transform on finite abelian groups, we study some arithmetic properties of indicators. We also give some applications to Hopf algebras.

Cited by 20 publications

References 26 publications

The pivotal cover and Frobenius–Schur indicators

The pivotal cover and Frobenius–Schur indicators

Signatures of broken parity and time-reversal symmetry in generalized string-net models

Lifting shadings on symmetrically self-dual subfactor planar algebras

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