Cyclic Homology of Hopf Comodule Algebras and Hopf Module Coalgebras

Abstract

In this paper we construct a cylindrical module A ♮ ℋ for an ℋ-comodule algebra A, where the antipode of the Hopf algebra ℋ is bijective. We show that the cyclic module associated to the diagonal of A ♮ ℋ is isomorphic with the cyclic module of the crossed product algebra A ⋊ ℋ. This enables us to derive a spectral sequence for the cyclic homology of the crossed product algebra. We also construct a cocylindrical module for Hopf module coalgebras and establish a similar spectral sequence to compute the cyclic cohomology of crossed product coalgebras.

Key Words:

• Cyclic homology
• Hopf algebras

Acknowledgment

We would like to thank the referee for pointing out a number of typographical errors in the original version of this paper.