Bahram Rangipour

A New Cyclic Module for Hopf Algebras

We define a new cyclic module, dual to the Connes-Moscovici cyclic module, for Hopf algebras, and give a characteristric map for the coaction 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

ABSTRACT 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.

Introduction to Hopf-Cyclic Cohomology

We review the recent progress in the study of cyclic cohomology in the presence of Hopf symmetry

SAYD modules over Lie-Hopf algebras

In this paper a general van Est type isomorphism is proved. 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 those modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is proved 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.

Generalized Coefficients for Hopf Cyclic Cohomology

A category of coefficients for Hopf cyclic cohomology is defined. It is shown that this category has two proper subcategories of which the smallest one is the known category of stable anti Yetter{Drinfeld modules. The middle subcategory is comprised of those coefficients which satisfy a generalized SAYD condition depending on both the Hopf algebra and the (co)algebra in question. Some examples are introduced to show that these three categories are different. It is shown that all components of Hopf cyclic cohomology work well with the new coefficients we have defined.

Constant and Equivariant Cyclic Cohomology

In this note, we prove that the constant and equivariant cyclic cohomology of algebras coincide. This shows that constant cyclic cohomology is rich and computable.