TITLE: Induced coaction on a vector space.
QUESTION [4 upvotes]: Let $H$ be a Hopf algebra and let $V$ be an $H$-module. Then we have an action $H \times V \to V$ given by $(h, v) \mapsto h.v$. This action induces an $H$-action on the dual vector space $V^*$ of $V$ which is given by $(h, \lambda) \mapsto h.\lambda$, where $(h.\lambda)(v) = \lambda(S(h).v)$, $S$ is the antipode in $H$. 
Suppose that $U$ is an $H$-comodule. Then we have a coaction $\delta: U \to H \otimes U$. Let $U^*$ be the dual vector space of $U$. Do we have some natural coaction $U^* \to H \otimes U^*$ induced by $\delta$? Thank you very much.

REPLY [3 votes]: When $U$ is finite dimensional the answer is yes.  Think of $U^\ast$ and $H \otimes U^\ast$ as $\hom(U, k)$ and $\hom(U, H)$ respectively.  Then the map $U^\ast \to H \otimes U^\ast$ is given by $\phi \mapsto (\mathrm{id}_H \overline\otimes\phi)\circ\delta$, where here $f \overline\otimes g$ is the map $a \otimes b \mapsto f(a)g(b)$.
{"set_name": "stack_exchange", "score": 4, "question_id": 1412182}