Hopf Algebra Equivariant Cyclic Homology and Cyclic Homology of Crossed Product Algebras

TL;DR

This paper develops a new framework for understanding cyclic homology of crossed product algebras using Hopf algebra symmetries, introducing a cylindrical module and spectral sequence analysis.

Contribution

It introduces the cylindrical module $A \natural \mathcal{H}$ and establishes isomorphisms with cyclic modules of crossed product algebras, providing new tools for cyclic homology computations.

Findings

Isomorphism between cyclic modules of crossed product and cylindrical module.

Spectral sequence approximation of cyclic homology for crossed products.

Interpretation of spectral sequence terms in the context of Hopf algebra actions.

Abstract

We introduce the cylindrical module $A \natural \mathcal{H}$, where $\mathcal{H}$ is a Hopf algebra and $A$ is a Hopf module algebra over $\mathcal{H}$. We show that there exists an isomorphism between $\mathsf{C}_{\bullet}(A^{op} \rtimes \mathcal{H}^{cop})$ the cyclic module of the crossed product algebra $A^{op} \rtimes \mathcal{H}^{cop} $, and $\Delta(A \natural \mathcal{H}) $, the cyclic module related to the diagonal of $A \natural \mathcal{H}$. If $S$, the antipode of $\mathcal{H}$, is invertible it follows that $\mathsf{C}_{\bullet}(A \rtimes \mathcal{H}) \simeq \Delta(A^{op} \natural \mathcal{H}^{cop})$. When $S$ is invertible, we approximate $HC_{\bullet}(A \rtimes \mathcal{H})$ by a spectral sequence and give an interpretation of $ \mathsf{E}^0, \mathsf{E}^1$ and $\mathsf{E}^2 $ terms of this spectral sequence.