Bivariant Hopf Cyclic Cohomology

Abstract

For module algebras and module coalgebras over an arbitrary bialgebra, we define two types of bivariant cyclic cohomology groups called bivariant Hopf cyclic cohomology and bivariant equivariant cyclic cohomology. These groups are defined through an extension of Connes' cyclic category Λ. We show that, in the case of module coalgebras, bivariant Hopf cyclic cohomology specializes to Hopf cyclic cohomology of Connes and Moscovici and its dual version by fixing either one of the variables as the ground field. We also prove an appropriate version of Morita invariance for both of these theories.

Key Words:

• Bivariant cyclic cohomology
• Conne's cyclic category
• Cyclic duality
• Hopf algebra
• Hopf-cyclic cohomology

2000 Mathematics Subject Classification:

• Primary 58B34, 16T99
• Secondary 19D55, 18G99

ACKNOWLEDGMENT

We would like to thank the referee for her/his careful reading of the text, and corrections and suggestions s/he made. The first author would like to thank Max Planck Institute in Bonn for their generous support and hospitality during which this work was finished.

Notes

Communicated by C. Cibils.