# Hopf Algebra Pairings and Module-Comodule-Equivalences

Let $\left< , \right> : G \times H \to {\bf C}$ be a dually pairing for two complex Hopf algebras $G$ and $H$. For any (left)-$G$-comodule $(V,\Delta_R)$, we can give $V$ the structure of a left $H$-module by defining $$G \times V \to V, ~~~~~~~~~ (h,v) \mapsto v_{(0)} \left< v_{(1)},S(h) \right> .$$ Now it is not clear to me how to go the other way, ie how to produce a $G$-comodule from a $H$-module.

For the case of $U_q[\frak{sl}_2]$, and the coordinate algebra $SL_p(2)$, we have such a correspondence between comodules, but the pairing between these two Hopf algebras is non-degenerate.

This, causes me to propose the conjecture: A dual pairing, for $G$ and $H$ two (complex) Hopf algebras, induces an equivalence of modules of $G$ and comodules of $H$, and vice-versa if, and only if, the pairing is non-degenerate.

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As far as I know, the reverse direction ($G$-comodule from $H$-module) requires finite-dimensionality or a similar hypothesis. And it's not very explicit -- it uses the isomorphism $\mathrm{Hom}\left(X,Y\otimes Z\right) \cong \mathrm{Hom}\left(Z^{\ast}\otimes X,Y\right)$ for any three vector spaces $X$, $Y$, $Z$ with $Z$ being finite-dimensional. One direction of this isomorphism can be directly written down; the other follows from $V^{\ast\ast}\cong V$. – darij grinberg Mar 22 '13 at 18:20