\sigma_{0} = \sigma_{n} \tau_{n+1}^2$ can be obtained from the other relations. From a (co)cyclic module, one can define Hochschild, cyclic and periodic cyclic (co)homology [@loday].

$H$-modules, $H$-comodules, and compatibility conditions. {#ydsection}
---------------------------------------------------------

Recall that the center of the monoidal category of left $H$-modules, ${{_H\mathcal{M}}}$, is equivalent to the category of left-right Yetter-Drinfeld modules. For finite dimensional Hopf algebras the center $\mathcal{Z}({{_H\mathcal{M}}})$ is also equivalent to the representations of the quantum double $_{D(H)}\mathcal{M}$. For details, we refer the reader to [@kassel]. We recall from [@majid; @sch] that for a Hopf algebra (or a bialgebra) $H$, a left $H$-module, right $H$-comodule $M$ is called a Yetter-Drinfeld module if $$\label{YD1}
  h{~\hspace{-3pt}^{^{\left(1\right)}}}m{~\hspace{-3pt}_{\left<0\right>}}\ot h{~\hspace{-3pt}^{^{\left(2\right)}}} m{~\hspace{-3pt}_{\left<1\right>}}=(h{~\hspace{-3pt}^{^{\left(2\right)}}} m){~\hspace{-3pt}_{\left<0\right>}}\ot (h{~\hspace{-3pt}^{^{\left(2\right)}}} m){~\hspace{-3pt}_{\left<1\right>}}h{~\hspace{-3pt}^{^{\left(1\right)}}}.$$ For Hopf algebras with an invertible antipode this is equivalent to $$\label{YD2}
  (hm){~\hspace{-3pt}_{\left<0\right>}}\ot (hm){~\hspace{-3pt}_{\left<1\right>}}= h{~\hspace{-3pt}^{^{\left(2\right)}}}m{~\hspace{-3pt}_{\left<0\right>}}\ot h{~\hspace{-3pt}^{^{\left(3\right)}}}m{~\hspace{-3pt}_{\left<1\right>}}S^{-1}(h{~\hspace{-3pt}^{^{\left(1\right)}}}).$$ The isomorphism giving the central structure of a left right YD module $M$ is $$\begin{aligned}
V\ot M&\xrightarrow\sim M\ot V\\v\ot m&\mapsto m{~\hspace{-3pt}_{\left<0\right>}}\ot m{~\hspace{-3pt}_{\left<1\right>}}v\end{aligned}$$ and the YD condition simply ensures that the map above is that of $H$-modules.

We recall from [@hkrs2] that a left-right anti Yetter-Drinfeld module $M$ over a Hopf algebra $H$ is a left $H$-module and a right $H$-comodule satisfying $$\label{AYD1}
   (hm){~\hspace{-3pt}_{\left<0\right>}}\ot (hm){~\hspace{-3pt}_{\left<1\right>}}=  h{~\hspace{-3pt}^{^{\left(2\right)}}}m{~\hspace{-3pt}_{\left<0\right>}}\ot h{~\hspace{-3pt}^{^{\left(3\right)}}}m{~\hspace{-3pt}_{\left<1\right>}}S(h{~\hspace{-3pt}^{^{\left(1\right)}}}).$$ We denote the category of 