Paragroupe d’Adrian Ocneanu et algèbre de Kac

David, Marie-Claude

doi:10.2140/pjm.1996.172.331

Cited by 27 publications

(16 citation statements)

References 10 publications

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“…Thus, θ is a von Neumann algebra isomorphism. (ii) If the inclusion N ⊂ M is irreducible, then B is a usual Kac algebra (i.e., a Hopf C * -algebra) and we recover the well-known result proved in [18], [10], and [3].…”

Section: Proposition 56s Is a Linear Anti-multiplicative And Anti-cosupporting

confidence: 77%

“…the corresponding Jones tower. It was announced by A. Ocneanu and was proved in [18], [3], [10] that if N ⊂ M is irreducible, i.e., such that N ′ ∩ M = C, then B = M ′ ∩ M 2 has a natural structure of a finite-dimensional Kac algebra and there is a canonical outer action of B on M 1 such that M = M B 1 , the fixed point subalgebra of M 1 with respect to this action, and M 2 is isomorphic to the crossed product M 1 >⊳B. The outerness condition is equivalent to the relative commutant M ′ 1 ∩ M 1 >⊳B being trivial (such actions are also called minimal).…”

Section: Introductionmentioning

confidence: 92%

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A Characterization of Depth 2 Subfactors of II1 Factors

Nikshych¹,

Vaînerman²

2000

Journal of Functional Analysis

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We characterize finite index depth 2 inclusions of type II 1 factors in terms of actions of weak Kac algebras and weak C*-Hopf algebras. If N/M/M 1 / M 2 / } } } is the Jones tower constructed from such an inclusion N/M, then B= M$ & M 2 has a natural structure of a weak C*-Hopf algebra and there is a minimal action of B on M 1 such that M is the fixed point subalgebra of M 1 and M 2 is isomorphic to the crossed product of M 1 and B. This extends the well-known results for irreducible depth 2 inclusions. Academic Press

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Section: Proposition 56s Is a Linear Anti-multiplicative And Anti-cosupporting

confidence: 77%

Section: Introductionmentioning

confidence: 92%

A Characterization of Depth 2 Subfactors of II1 Factors

Nikshych¹,

Vaînerman²

2000

Journal of Functional Analysis

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“…The coinvolution, or antipode, of a Kac algebra constructed from a depth 2 irreducible inclusion coincides with j 2 (voir [Dav96]). From [Kas95, III.3.2], the antipode of a C * -algèbre de Hopf is uniquely determined by the rest of the structure.…”

Section: I) (4) the Indices Satisfy The Following Relationsmentioning

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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“…It is known that the finite dimensional Kac algebras are characterized as special paragroups. (See David [1], Longo [6], Szymanski [15], and also Yamanouchi [17]. )…”

Section: Introductionmentioning

confidence: 99%