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## Coaction on a finite dual of a Hopf algebra

If $H$ is a central hopf subalgebra of a Hopf algebra $A$. Let $\phi:A\rightarrow A\otimes H$ be a coaction of $H$ on $A$. When does $A^\circ$ becomes a $H^{*}$-submodule i.e. is $A^\circ$ an $H$-subcomodule? Basically I want to realize given a coaction of $H$ on $A$, does it imply a coaction of $H$ on the finite dual of $A$ as well: $$A^\circ\rightarrow A^\circ\otimes H \hspace{8 pt}?$$

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