Cyclic Homology of Strong Smash Product Algebras

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.