A van Est Isomorphism for Bicrossed Product Hopf Algebras

Rangipour, Bahram; Sütlü, Serkan

doi:10.1007/s00220-012-1452-9

Cited by 10 publications

(18 citation statements)

References 25 publications

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“…Let now g be a finite-dimensional Lie algebra and let F be a g-Hopf algebra (cf. [23]), on which g coacts via g : g → g ⊗ F . The modular character of δ : g → C, δ(X) = Trace(ad X ), X ∈ g, extends to a character of U(g).…”

Section: )mentioning

confidence: 99%

“…[18,19]), and therefore it also is a repository of the universal Hopf cyclic Chern classes. The proof of this isomorphism is achieved by supplementing our earlier techniques with those in [23]. By a construction parallel to that in [21], we then realize these classes in terms of concrete geometric cocycles, in the spirit of the Chern-Weil theory.…”

Section: Introductionmentioning

confidence: 99%

“…To equip the algebra K n itself with a Hopf algebra structure, we shall check that the Lie algebra V generated by the X ℓ 's together with the copposite Hopf algebra F K = K ab cop form a Lie-Hopf pair in the sense of [23]. It will follow that the universal enveloping algebra U(V ) of V together with F K form a matched pair of Hopf algebras (cf.…”

Section: Introductionmentioning

confidence: 99%

“…We need to verify that the action ⊲ and the coaction satisfy the conditions required for a Lie-Hopf pair (cf. [23]). First we should check that for any g ∈ F K and any X ∈ V one has ∆(X ⊲ g) = X • ∆(g) = g(1) ⊗ X ⊲ g(2) + X <0> ⊲ g(1) ⊗ X <1> g (2).…”

Section: Introductionmentioning

confidence: 99%

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Hopf algebras and universal Chern classes

Moscovici¹,

Rangipour²

2017

J. Noncommut. Geom.

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We construct a variant K n of the Hopf algebra H n , which acts directly on the noncommutative model for the space of leaves of a general foliation rather than on its frame bundle. We prove that the Hopf cyclic cohomology of K n is isomorphic to that of the pair (H n , gl n ) and thus consists of the universal Hopf cyclic Chern classes. We also realize these classes in terms of geometric cocycles.

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Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hopf algebras and universal Chern classes

Moscovici¹,

Rangipour²

2017

J. Noncommut. Geom.

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“…Next, in [29], further examples of MPI's were constructed for bicrossproduct Hopf algebras associated to Lie algebras. These MPI's were then upgraded into nontrivial SAYD modules in [28].…”

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confidence: 99%

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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SAYD Modules over Lie-Hopf Algebras

Rangipour

Sütlü

2012

Commun. Math. Phys.

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In this paper a general van Est type isomorphism is established. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and SAYD modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is found at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes-Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate the whole theory on this example. Finally explicit representative cocycles of the cohomology classes for this example are calculated.

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A van Est Isomorphism for Bicrossed Product Hopf Algebras

Cited by 10 publications

References 25 publications

Hopf algebras and universal Chern classes

Hopf algebras and universal Chern classes

Cyclic Homology and Quantum Orbits

SAYD Modules over Lie-Hopf Algebras

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