Equivariant Hopf Galois extensions and Hopf cyclic cohomology

Hassanzadeh, Mohammad; Rangipour, Bahram

doi:10.4171/jncg/110

Cited by 7 publications

(13 citation statements)

References 21 publications

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“…Let us next recall the cyclic homology of module corings with SAYD coefficients from [3,19]. Let B be a right × A -Hopf algebra, C an A-coring as well as a right B-module coring, i.e.…”

Section: 2mentioning

confidence: 99%

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Cyclic Homology and Quantum Orbits

Maszczyk

Sütlü²

2015

SIGMA

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Abstract. A natural isomorphism between the cyclic object computing the relative cyclic homology of a homogeneous quotient-coalgebra-Galois extension, and the cyclic object computing the cyclic homology of a Galois coalgebra with SAYD coefficients is presented. The isomorphism can be viewed as the cyclic-homological counterpart of the Takeuchi-Galois correspondence between the left coideal subalgebras and the quotient right module coalgebras of a Hopf algebra. A spectral sequence generalizing the classical computation of Hochschild homology of a Hopf algebra to the case of arbitrary homogeneous quotient-coalgebra-Galois extensions is constructed. A Pontryagin type self-duality of the Takeuchi-Galois correspondence is combined with the cyclic duality of Connes in order to obtain dual results on the invariant cyclic homology, with SAYD coefficients, of algebras of invariants in homogeneous quotient-coalgebra-Galois extensions. The relation of this dual result with the Chern character, Frobenius reciprocity, and inertia phenomena in the local Langlands program, the Chen-Ruan-Brylinski-Nistor orbifold cohomology and the Clifford theory is discussed.

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“…Let us next recall the cyclic homology of module corings with SAYD coefficients from [3,19]. Let B be a right × A -Hopf algebra, C an A-coring as well as a right B-module coring, i.e.…”

Section: 2mentioning

confidence: 99%

“…(4. 19) in (A ⊗ A) B ⊗ H ⊗ H. The latter implies h [2] (2) a(0)h [1] (2) ⊗ h(3) a(1)S(h(1)) = h [2] (0) a(0)h [1] (0) ⊗ h [2] (1) a(1) h [1] (1) (4.20) in A B ⊗ H, which amounts to the fact that the map (4.17) of SAYD modules is also H-colinear.…”

Section: 2mentioning

confidence: 99%

Cyclic Homology and Quantum Orbits

Maszczyk

Sütlü²

2015

SIGMA

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“…In the extended version, the cyclic cohomology of Lu's Hopf algebroid (which is defined in [CM01] and [Ko]) and the cyclic cohomology of Khalkhali-Rangipour's para-Hopf algebroid [KR3] are defined with trivial coefficients. Cyclic cohomology theory with generalized coefficients for extended versions of Hopf algebras was first defined in [BS] for ×-Hopf algebras and later was generalized in [HR1], [HR2] and [KK]. The authors of [BS2] developed a categorial approach to find cyclic objects for ×-Hopf algebras.…”

Section: Introductionmentioning

confidence: 99%

“…We note that in the bialgebroid structure we have equal source and target maps s = t : C −→ K given by c −→ c1 K . The following lemma [HR1], which implies some properties of the map ν will be used in the next subsection.…”

Section: Introductionmentioning

confidence: 99%

On Cyclic Cohomology of ×-Hopf algebras

Hassanzadeh¹

2014

J K-Theor

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In this paper we study the cyclic cohomology of certain -Hopf algebras: universal enveloping algebras, quantum algebraic tori, the Connes-Moscovici -Hopf algebroids and the Kadison bialgebroids. Introducing their stable anti Yetter-Drinfeld modules and cocyclic modules, we compute their cyclic cohomology. Furthermore, we provide a pairing for the cyclic cohomology of -Hopf algebras which generalizes the Connes-Moscovici characteristic map to -Hopf algebras. This enables us to transfer the -Hopf algebra cyclic cocycles to algebra cyclic cocycles.

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“…By [Has,Lemma 3.5] the map κ is a right H-module map and therefore it is a total cointegral. Therefore we obtain the following lemma.…”

mentioning

confidence: 99%

On representation theory of total (co)integrals

Hassanzadeh

2016

J. Algebra Appl.

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Abstract. In this paper, we show that total integrals and cointegrals are new sources of stable anti Yetter-Drinfeld modules. We explicitly show that how special types of total (co)integrals can be used to provide both (stable) anti Yetter-Drinfeld and YetterDrinfeld modules. We use these modules to classify total (co)integrals and (cleft) Hopf Galois (co)extensions for some examples of the Connes-Moscovici Hopf algebra, universal enveloping algebras and polynomial algebras.

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Equivariant Hopf Galois extensions and Hopf cyclic cohomology

Cited by 7 publications

References 21 publications

Cyclic Homology and Quantum Orbits

Cyclic Homology and Quantum Orbits

On Cyclic Cohomology of ×-Hopf algebras

On representation theory of total (co)integrals

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