A Characterization of Depth 2 Subfactors of II1 Factors

Nikshych, Dmitri; Vaînerman, Leonid

doi:10.1006/jfan.1999.3522

Cited by 58 publications

(95 citation statements)

References 19 publications

(28 reference statements)

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“…We recall that the inclusion is irreducible if and only if B is a Kac algebra. We refer to [NV00b] for historical notes on the problem and references to analogous results.…”

Section: ])mentioning

confidence: 99%

Exploration of Finite-Dimensional Kac Algebras and Lattices of Intermediate Subfactors of Irreducible Inclusions

David

Thiéry

2011

J. Algebra Appl.

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Abstract. We study the four infinite families KA(n), KB(n), KD(n), KQ(n) of finite dimensional Hopf (in fact Kac) algebras constructed respectively by A. Masuoka and L. Vainerman: isomorphisms, automorphism groups, self-duality, lattices of coideal subalgebras. We reduce the study to KD(n) by proving that the others are isomorphic to KD(n), its dual, or an index 2 subalgebra of KD(2n). We derive many examples of lattices of intermediate subfactors of the inclusions of depth 2 associated to those Kac algebras, as well as the corresponding principal graphs, which is the original motivation.Along the way, we extend some general results on the Galois correspondence for depth 2 inclusions, and develop some tools and algorithms for the study of twisted group algebras and their lattices of coideal subalgebras. This research was driven by heavy computer exploration, whose tools and methodology we describe.

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“…We recall that the inclusion is irreducible if and only if B is a Kac algebra. We refer to [NV00b] for historical notes on the problem and references to analogous results.…”

Section: ])mentioning

confidence: 99%

Exploration of Finite-Dimensional Kac Algebras and Lattices of Intermediate Subfactors of Irreducible Inclusions

David

Thiéry

2011

J. Algebra Appl.

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“…This link between quantum groups and irreducible inclusions of depth 2 goes back to Ocneanu and was also generalized to reducible inclusions, yielding quantum groupoids, see e.g. [25].…”

Section: Introductionmentioning

confidence: 82%

Strictly Outer Actions of Groups and Quantum Groups

Vaes

2005

Journal Fur Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. An action of a locally compact group or quantum group on a factor is said to be strictly outer when the relative commutant of the factor in the crossed product is trivial. We show that all locally compact quantum groups can act strictly outerly on a free Araki-Woods factor and that all locally compact groups can act strictly outerly on the hyperfinite II 1 factor. We define a kind of Connes' T invariant for locally compact quantum groups and prove a link with the possibility of acting strictly outerly on a factor with a given T invariant. Necessary and sufficient conditions for the existence of strictly outer actions of compact Kac algebras on the hyperfinite II 1 factor are given.

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“…This appears to be a natural property, since it is satisfied by dynamical quantum groups [4] and weak Hopf algebras arising as symmetries of Jones-von Neumann subfactors [13], [14].…”

Section: Minimal Weak Hopf Algebrasmentioning

confidence: 99%

On the Structure of Weak Hopf Algebras

Nikshych¹

2002

Advances in Mathematics

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Abstract. We study the group of group-like elements of a weak Hopf algebra and derive an analogue of Radford's formula for the fourth power of the antipode S, which implies that the antipode has a finite order modulo a trivial automorphism. We find a sufficient condition in terms of Tr(S 2 ) for a weak Hopf algebra to be semisimple, discuss relation between semisimplicity and cosemisimplicity, and apply our results to show that a dynamical twisting deformation of a semisimple Hopf algebra is cosemisimple.

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A Characterization of Depth 2 Subfactors of II1 Factors

Cited by 58 publications

References 19 publications

Exploration of Finite-Dimensional Kac Algebras and Lattices of Intermediate Subfactors of Irreducible Inclusions

Exploration of Finite-Dimensional Kac Algebras and Lattices of Intermediate Subfactors of Irreducible Inclusions

Strictly Outer Actions of Groups and Quantum Groups

On the Structure of Weak Hopf Algebras

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