Relations between representations and corepresentations of dually paired Hopf algebras It is well known that if two Hopf algebras $A, B$ are dually paired and $\phi$ is a corepresentation of $A$ then it canonically induces a representation $R_\phi$ of $B$. I have a few "converse" questions:
1) Is it known when a representation $\psi$ of $B$ is canonically induced from a corepresentation of $A$, i.e., $\psi = R_\phi$ for $\phi$ some corep of $A$?
2) Is it known when a representation $\psi$ of $B$ canonically induces a corepresentation $C_\psi$ of $A$?
3) If $\psi$ is a reprensetation of $B$ which canonically induces a corepresentation $C_\psi$ of $A$, is the canonically induced reprensetation $R_{C_\psi}$ of $B$ isomorphic to $\psi$?
What I am looking for here are necessary, sufficient or necessary and sufficient conditions for the questions, but of course I would be totally satisfied with partial answers or references.
I am not completely sure, but it does seem that there is a "complete" duality between tempered representations of a semi-simple Lie group $G$ and some nice weight representations of the universal enveloping algebra $U(\mathfrak{g})$ of its Lie algebra.
 A: I'm not sure if this is what you're asking for. Let us assume $V$ to be a representation of an algebra de hopf $H$ then you have the representation determined by an action $\rho:\,A\otimes V\longrightarrow V$ such that $$\rho{\scriptstyle \circ}\left(\eta\otimes id_{V}\right) =id_{V},$$
$$\rho{\scriptstyle \circ}\left(id_{A}\otimes\rho\right) =\rho{\scriptstyle \circ}\left(\mu\otimes id_{V}\right)$$
Similarly for the corepresentation you need a co-action $\delta:\,V\longrightarrow C\otimes V $ such that $$\left(\varepsilon\otimes id_{V}\right){\scriptstyle \circ}\delta =id_{V},$$
$$\left(id_{C}\otimes\delta\right){\scriptstyle \circ}\delta =\left(\bigtriangleup\otimes id_{V}\right){\scriptstyle \circ}\delta.$$
Now, if you have a co-module of $H$ with action $\delta$ then you can define an action on the dual $\rho:\,H^{*}\otimes V^{*}\longrightarrow V^{*}$ defining
$$\rho\left(\xi\otimes f\right)\left(v\right):=\left(\xi\otimes f\right)\delta\left(v\right),$$
Now, if you have a Hopf pairing between $U$ and $H$, i.e. $\left\langle u,\,x\right\rangle$, then you have that   $\left\langle u,\,.\right\rangle:U\longrightarrow H^{*}$ is a morphism of algebra. So you use this morphism to extend the action from $H^*$ to $U$ and then you can define an action from $\rho:\,U\otimes V^{*}\longrightarrow V^{*}$. If you want to find it explicitly is $$\rho\left(u\otimes f\right)\left(v\right):=\left(\left\langle u,\,.\right\rangle\otimes f\right)\delta\left(v\right).$$
