Frobenius algebras in tensor categories and bimodule extensions

Yamagami, Shigeru

doi:10.1090/fic/043/27

Cited by 23 publications

(17 citation statements)

References 11 publications

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“…If A is simple, the quantum dimension dim A ( · ) in C A|A is given in terms of the quantum dimension dim( · ) of C by dim A (X) = dim(X)/dim(A) (this follows for example from writing out the definition of dim A ( · ) and using lemma 4.1 below). (iii) For a modular tensor category C one can consider the 2-category Frob C whose objects are symmetric special Frobenius algebras in C (compare also [36,37,35] and section 3 of [38]). The categories Frob C (A, B) (whose objects are the 1-morphisms A → B) are the categories C A|B of A-B-bimodules and the 2-morphisms are morphisms of bimodules.…”

Section: Proposition 24mentioning

confidence: 99%

Duality and defects in rational conformal field theory

et al. 2007

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We study topological defect lines in two-dimensional rational conformal field theory. Continuous variation of the location of such a defect does not change the value of a correlator. Defects separating different phases of local CFTs with the same chiral symmetry are included in our discussion. We show how the resulting onedimensional phase boundaries can be used to extract symmetries and order-disorder dualities of the CFT. The case of central charge c = 4/5, for which there are two inequivalent world sheet phases corresponding to the tetra-critical Ising model and the critical three-states Potts model, is treated as an illustrative example.

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Section: Proposition 24mentioning

confidence: 99%

Duality and defects in rational conformal field theory

et al. 2007

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“…Remark 2.19 Note that in the literature the generalization of Q-systems to the nonalgebraic context is typically that of a Frobenius Algebra [Müg03,Yam04]. However, in context of simple algebras, the Frobenius trace just comes from the (unique up to rescaling) splitting of the unit morphism.…”

Section: Subfactorsmentioning

confidence: 99%

Quantum Subgroups of the Haagerup Fusion Categories

Grossman

Snyder

2012

Commun. Math. Phys.

105

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We answer three related questions concerning the Haagerup subfactor and its even parts, the Haagerup fusion categories. Namely we find all simple module categories over each of the Haagerup fusion categories (in other words, we find the "quantum subgroups" in the sense of Ocneanu), we find all subfactors whose principal even part is one of the Haagerup fusion categories, and we compute the Brauer-Picard groupoid of Morita equivalences of the Haagerup fusion categories. In addition to the two even parts of the Haagerup subfactor, there is exactly one more fusion category which is Morita equivalent to each of them. This third fusion category has six simple objects and the same fusion rules as one of the even parts of the Haagerup subfactor, but has not previously appeared in the literature. We also find the full lattice of intermediate subfactors for every subfactor whose even part is one of these three fusion categories, and we discuss how our results generalize to Izumi subfactors.

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“…The corresponding compression of O(P • ) is the Doplicher-Roberts algebra O ρ , where ρ corresponds the strand in Pro(P • ), the rigid C * -tensor category of projections of P • [MPS10,Yam12,BHP12]. Hence we recover the main result of [MRS92] on compressing O Γρ to obtain O ρ .…”

Section: Introductionmentioning

confidence: 66%

“…Moreover, the fusion graph Γ ρ corresponds to the principal graph Γ. Also see [Yam12] for more details.…”

Section: Planar Algebrasmentioning

confidence: 99%

C-algebras from planar algebras I: Canonical C-algebras associated to a planar algebra

Hartglass

Penneys

2016

Trans. Amer. Math. Soc.

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From a planar algebra, we give a functorial construction to produce numerous associated C ∗ ^* -algebras. Our main construction is a Hilbert C ∗ ^* -bimodule with a canonical real subspace which produces Pimsner-Toeplitz, Cuntz-Pimsner, and generalized free semicircular C ∗ ^* -algebras. By compressing this system, we obtain various canonical C ∗ ^* -algebras, including Doplicher-Roberts algebras, Guionnet-Jones-Shlyakhtenko algebras, universal (Toeplitz-) Cuntz-Krieger algebras, and the newly introduced free graph algebras. This is the first article in a series studying canonical C ∗ ^* -algebras associated to a planar algebra.

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Frobenius algebras in tensor categories and bimodule extensions

Cited by 23 publications

References 11 publications

Duality and defects in rational conformal field theory

Duality and defects in rational conformal field theory

Quantum Subgroups of the Haagerup Fusion Categories

C-algebras from planar algebras I: Canonical C-algebras associated to a planar algebra

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Frobenius algebras in tensor categories and bimodule extensions

Cited by 23 publications

References 11 publications

Duality and defects in rational conformal field theory

Duality and defects in rational conformal field theory

Quantum Subgroups of the Haagerup Fusion Categories

C*-algebras from planar algebras I: Canonical C*-algebras associated to a planar algebra

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C-algebras from planar algebras I: Canonical C-algebras associated to a planar algebra