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Let $H$ be a Hopf algebra and $A$ an algebra. Let $H^*$ be the dual Hopf algebra of $H$. Then by Proposition 1.6.11 in the book Foundations of Quantum Group Theory by Shahn Majid, $A$ is a right $H$-comodule algebra if and only if $A$ is a left $H^*$-module algebra.

If the right coaction of $H$ on $A$ is given by $a \mapsto \sum a_{(0)} \otimes a_{(1)}$, then the left $H^*$-action on $A$ is given by $f \mapsto \sum a_{(0)} \langle f, a_{(1)} \rangle$, where $\langle \cdot, \cdot \rangle$ is the pairing between $H^*$ and $H$.

Can "right $H$-comodule algebra" in the above proposition be replaced by "left $H$-comodule algebra"? Is the following also true?

$A$ is a left $H$-comodule algebra if and only if $A$ is a left $H^*$-module algebra.

Thank you very much.

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    $\begingroup$ Yes, of course. Apply the theorem to $H_{op}$, which is the same as $H$ but with product $a \cdot_{op} b = ba$ and coproduct $\Delta_{op} = \tau \Delta$, where $\tau : H^{\otimes 2} \to H^{\otimes 2}$ exchanges the two factors. $\endgroup$
    – Najib Idrissi
    Commented Mar 6, 2018 at 15:12

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