Cyclic Homology of Strong Smash Product Algebras

TL;DR

This paper develops a new approach to compute the cyclic homology of strong smash product algebras using cylindrical modules and spectral sequences, with applications to specific algebra examples like Pareigis' Hopf algebra.

Contribution

It introduces a cylindrical module construction and spectral sequence framework for cyclic homology of strong smash product algebras, providing explicit calculations including Pareigis' Hopf algebra.

Findings

Constructed a cylindrical module $A\natural B$ for strong smash product algebras.

Proved the diagonal cyclic module is isomorphic to the algebra's cyclic module.

Established a spectral sequence converging to the cyclic homology.

Abstract

For any strong smash product algebra $A\#_{_R}B$ of two algebras $A$ and $B$ with a bijective morphism $R$ mapping from $B\ot A$ to $A\ot B$, we construct a cylindrical module $A\natural B$ whose diagonal cyclic module $\Delta_{\bullet}(A\natural B)$ is graphically proven to be isomorphic to $C_{\bullet}(A\#_{_R}B)$ the cyclic module of the algebra. A spectral sequence is established to converge to the cyclic homology of $A\#_{_R}B$. Examples are provided to show how our results work. Particularly, the cyclic homology of the Pareigis' Hopf algebra is obtained in the way.