Finite Index Subfactors and Hopf Algebra Crossed Products

Szymański, Wojciech

doi:10.2307/2159890

Cited by 35 publications

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“…According to [16] there is a 12 dimensional Hopf- * -algebra A such that R G ⊂ F is equivalent to R A ⊂ R where A acts on R by an outer action. A is neither commutative nor cocommutative as in the principal and in the dual principal graph there are even vertices which are connected with the unique odd vertex by more than one edge.…”

Section: Examplesmentioning

confidence: 99%

II₁-subfactors associated with the C^∗-tensor category of a finite group

Schaflitzel¹

1998

Pacific J. Math.

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We determine the subfactors N ⊂ R of the hyperfinite II 1 -factor R with finite index for which the C * -tensor category of the associated (N, N )-bimodules is equivalent to the C * -tensor category U G of all unitary finite dimensional representations of a given finite group G. It turns out that every subfactor of that kind is isomorphic to a subfactor R G ⊂ (R ⊗ L(C r )) H , where R G is the fixed point algebra under an outer action α of G, H is a subgroup of G, ψ : H −→ U(C R ) is a unitary finite dimensional projective representation of H satisfying a certain additional condition and (R ⊗ L(C r )) H is the fixed point algebra under the action α | H ⊗ Ad ψ of H on R ⊗ L(C r ).

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Section: Examplesmentioning

confidence: 99%

II₁-subfactors associated with the C^∗-tensor category of a finite group

Schaflitzel¹

1998

Pacific J. Math.

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“…If A and B are factors then B is isomorphic to a crossed product of A by an (outer) action of a finite dimensional Kac (C * -Hopf) algebra ( [8], [13]). Izumi showed in [2] that if A and B are unital simple C * -algebras then there is an action of a finite dimensional C * -Hopf algebra H on B such that A is the fixed point C * -subalgebra B H .…”

Section: Introductionmentioning

confidence: 99%

Inclusions of unital $C^$-algebras of index-finite type with depth 2 induced by saturated actions of finite dimensional $C^$-Hopf algebras

Kodaka¹,

Teruya²

2009

MATH. SCAND.

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Let B be a unital C * -algebra and H a finite dimensional C * -Hopf algebra with its dual C * -Hopf algebra H 0 . We suppose that there is a saturated action of H on B and we denote by A its fixed point C * -subalgebra of B. Let E be the canonical conditional expectation from B onto A. In the present paper, we shall give a necessary and sufficient condition that there are a weak action of H 0 on A and a unitary cocycle σ of H 0 ⊗ H 0 to A satisfying that there is an isomorphism π of A σ H 0 onto B, which is the twisted crossed product of A by the weak action of H 0 on A and the unitary cocycle σ , such that F = E • π, where F is the canonical conditional expectation from A σ H 0 onto A.

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“…It is well understood now that Hopf algebras and weak Hopf algebras arise as non-commutative symmetries of Jones towers of certain finite index inclusions of topological algebras over the complex numbers. For a finite index von Neumann subfactor N ⊆ M it was shown by Szymański [26] that the depth 2 condition for the associated tower of centralizers C M N ⊆ C M 1 N ⊆ C M 2 M ⊆ · · · is equivalent to A = C M 1 N having a natural structure of a Hopf C * -algebra, if C M N = 1 In the general case where C M N ⊇ 1, it was shown by Vainerman and the second author [18] that the depth 2 condition is equivalent to A being a weak Hopf C * -algebra. In both cases, A acts on M in such a way that N = M A and M 1 ∼ = M#A; moreover, B = C M 2 M is naturally identified with the (weak) Hopf C * -algebra dual to A.…”

Section: Introductionmentioning

confidence: 99%

Frobenius Extensions and Weak Hopf Algebras

Kadison

Nikshych

2001

Journal of Algebra

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We study a symmetric Markov extension of k-algebras N → M, a certain kind of Frobenius extension with conditional expectation that is tracial on the centralizer and dual bases with a separability property. We place a depth two condition on this extension, which is essentially the requirement that the Jones tower N → M → M 1 → M 2 can be obtained by taking relative tensor products with centralizers A = C M 1 N and B = C M 2 M . Under this condition, we prove that N → M is the invariant subalgebra pair of a weak Hopf algebra action by A, i.e., that N = M A . The endomorphism algebra M 1 = End N M is shown to be isomorphic to the smash product algebra M#A.

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Finite Index Subfactors and Hopf Algebra Crossed Products

Cited by 35 publications

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II₁-subfactors associated with the C^∗-tensor category of a finite group

II₁-subfactors associated with the C^∗-tensor category of a finite group

Inclusions of unital $C^$-algebras of index-finite type with depth 2 induced by saturated actions of finite dimensional $C^$-Hopf algebras

Frobenius Extensions and Weak Hopf Algebras

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Finite Index Subfactors and Hopf Algebra Crossed Products

Cited by 35 publications

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II1-subfactors associated with the C∗-tensor category of a finite group

II1-subfactors associated with the C∗-tensor category of a finite group

Inclusions of unital $C^*$-algebras of index-finite type with depth 2 induced by saturated actions of finite dimensional $C^*$-Hopf algebras

Frobenius Extensions and Weak Hopf Algebras

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Product

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II₁-subfactors associated with the C^∗-tensor category of a finite group

II₁-subfactors associated with the C^∗-tensor category of a finite group

Inclusions of unital $C^$-algebras of index-finite type with depth 2 induced by saturated actions of finite dimensional $C^$-Hopf algebras