Bivariant Hopf Cyclic Cohomology (original) (raw)

K T ] 1 J ul 2 00 3 Hopf-cyclic homology and cohomology with coefficients

2003

Following the idea of an invariant differential complex, we construct general-type cyclic modules that provide the common denominator of known cyclic theories. The cyclicity of these modules is governed by Hopfalgebraic structures. We prove that the existence of a cyclic operator forces a modification of the Yetter-Drinfeld compatibility condition leading to the concept of a stable anti-Yetter-Drinfeld module. This module plays the role of the space of coefficients in the thus obtained cyclic cohomology of module algebras and coalgebras, and the cyclic homology and cohomology of comodule algebras. Along the lines of Connes and Moscovici, we show that there is a pairing between the cyclic cohomology of a module coalgebra acting on a module algebra and closed 0-cocycles on the latter. The pairing takes values in the usual cyclic cohomology of the algebra. Similarly, we argue that there is an analogous pairing between closed 0-cocycles of a module coalgebra and the cyclic cohomology of...

Cyclic cohomology and Hopf algebras

Letters in Mathematical Physics, 1999

We associate canonically a cyclic module to any Hopf algebra endowed with a modular pair in involution, consisting of a group-like element and a character. This provides the key construction for allowing the extension of cyclic cohomology to Hopf algebras in the ...

On the Cyclic Cohomology of Extended Hopf Algebras

2001

We introduce the concept of {\it extended Hopf algebras} and consider their cyclic cohomology in the spirit of Connes-Moscovici. Extended Hopf algebras are closely related to, but different from, Hopf algebroids. Their definition is motivated by attempting to define cyclic cohomology of Hopf algebroids in general. Many of Hopf algebra like structures, including the Connes-Moscovici algebra mathcalHFM\mathcal{H}_{FM}mathcalHFM are extended Hopf algebras. We show that the cyclic cohomology of the extended Hopf algebra U(L,R)U(L,R)U(L,R) naturally associated to a Lie-Rinehart algebra (L,R)(L,R)(L,R) coincides with the homology of (L,R)(L,R)(L,R). We also give some other examples of extended Hopf algebras and their cyclic cohomology.

Hopf-cyclic homology and cohomology with coefficients

Comptes Rendus Mathematique, 2004

Following the idea of an invariant differential complex, we construct general-type cyclic modules that provide the common denominator of known cyclic theories. The cyclicity of these modules is governed by Hopfalgebraic structures. We prove that the existence of a cyclic operator forces a modification of the Yetter-Drinfeld compatibility condition leading to the concept of a stable anti-Yetter-Drinfeld module. This module plays the role of the space of coefficients in the thus obtained cyclic cohomology of module algebras and coalgebras, and the cyclic homology and cohomology of comodule algebras. Along the lines of Connes and Moscovici, we show that there is a pairing between the cyclic cohomology of a module coalgebra acting on a module algebra and closed 0-cocycles on the latter. The pairing takes values in the usual cyclic cohomology of the algebra. Similarly, we argue that there is an analogous pairing between closed 0-cocycles of a module coalgebra and the cyclic cohomology of a module algebra.

Lie-Hopf algebras and their Hopf cyclic cohomology

2010

The correspondence between Lie algebras, Lie groups, and algebraic groups, on one side and commutative Hopf algebras on the other side are known for a long time by works of Hochschild-Mostow and others. We extend this correspondence by associating a noncommutative noncocommutative Hopf algebra to any matched pair of Lie algebras, Lie groups, and affine algebraic groups. We canonically associate a modular pair in involution to any of these Hopf algebras. More precisely, to any locally finite representation of a matched pair object as above we associate a SAYD module to the corresponding Hopf algebra. At the end, we compute the Hopf cyclic cohomology of the associated Hopf algebra with coefficients in the aforementioned SAYD module in terms of Lie algebra cohomology of the Lie algebra associated to the matched pair object relative to an appropriate Levi subalgebra with coefficients induced by the original representation.

A New Cyclic Module for Hopf Algebras

K-Theory, 2000

We define a new cyclic module, dual to the Connes-Moscovici cocyclic module, for Hopf algebras, and give a characteristic map for coactions of Hopf algebras. We also compute the resulting cyclic homology for cocommutative Hopf algebras, and some quantum groups.

A Note on Cyclic Duality and Hopf Algebras

Communications in Algebra, 2005

We show that various cyclic and cocyclic modules attached to Hopf algebras and Hopf modules are related to each other via Connes' duality isomorphism for the cyclic category. * PIMS postdoctoral fellow