Noncommutative differential geometry on crossed product algebras

We provide a differential structure on arbitrary cleft extensions $B:=A^{\mathrm{c}\mathrm{o}H}\subseteq A$ for an $H$-comodule algebra $A$. This is achieved by constructing a covariant calculus on the corresponding crossed product algebra $B{\mathrm{\#}}_{\sigma }H$ from the data of a bicovariant calculus on the structure Hopf algebra $H$ and a calculus on the base algebra $B$, which is compatible with the $2$-cocycle and measure of the crossed product. The result is a quantum principal bundle with canonical strong connection and we describe the induced bimodule covariant derivatives on associated bundles of the crossed product. It is proven that connections of the quantum principal bundle are in bijection with connection $1$-forms. All results specialize to trivial extensions and smash product algebras $B\mathrm{\#}H$ and we give a characterization of the smash product calculus in terms of the differentials of the cleaving map $j\text{\hspace{0.17em}⁣}:H\to A$ and the inclusion $B\hookrightarrow A$. The construction is exemplified for pointed Hopf algebras. In particular, the case of Radford Hopf algebras $H_{(r,n,q)}$ is spelled out in detail.