. Furthermore, $$\Phi^{-1}(m\ot v)=S(m{~\hspace{-3pt}_{\left<1\right>}})v\ot m{~\hspace{-3pt}_{\left<0\right>}}.$$ We have arrived at the following theorem:

\[ydcenter\] For a Hopf algebra $H$ with an invertible antipode and $i\in\mathbb{Z}$,

$${\mathcal{Z}}_{{{_H\mathcal{M}}}}{{^{\#^i}_H\mathcal{M}}}\simeq {_H\mathcal{YD}}^H_{-i}.$$

If we consider the action of $\mathbb{Z}$ on ${{_H\mathcal{M}}}$ via $\#$, then ${{_H\mathcal{M}}}\rtimes\mathbb{Z}$ is a $\mathbb{Z}$ graded monoidal category. If we write ${{_H\mathcal{M}}}\rtimes\mathbb{Z}=\bigoplus_i{\mathcal{M}}_i$ then ${\mathcal{M}}_i={{_H\mathcal{M}}}^{\#^i}$ as an ${{_H\mathcal{M}}}$ bimodule category and $\bigoplus_i{_H\mathcal{YD}}^H_{i}$ is a $\mathbb{Z}$-equivariant $\mathbb{Z}$-braided monoidal category.

Stability. {#stabilitysection}
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Recall that a left $H$ module and right $H$ comodule $M$ is called stable if $m{~\hspace{-3pt}_{\left<1\right>}}m{~\hspace{-3pt}_{\left<0\right>}}=m$. We will need a slightly more general notion for the covariant theory. The classical stability will be precisely correct for the contravariant version.

Let $i\in\mathbb{Z}$. A left $H$ module and right $H$ comodule $M$ is called $i$-stable if $S^{2i}(m{~\hspace{-3pt}_{\left<1\right>}})m{~\hspace{-3pt}_{\left<0\right>}}=m$.

Thus the usual stability is now $0$-stability. The following lemma shows what happens to the odd powers of the antipode.

\[stability\] For a left $H$ module and right $H$ comodule $M$ we have $$S^{2i}(m{~\hspace{-3pt}_{\left<1\right>}})m{~\hspace{-3pt}_{\left<0\right>}}=m\quad\Leftrightarrow\quad S^{2i-1}(m{~\hspace{-3pt}_{\left<1\right>}})m{~\hspace{-3pt}_{\left<0\right>}}=m.$$

This is a direct computation, however in the instances where we see it, a more conceptual explanation can be found in terms of the $\tau_0$ map and its inverse that play a key role in our more conceptual understanding of stability.

“$\Rightarrow$" $$\begin{aligned}
