Internal Homs in Modules over a Hopf Algebra Given a Hopf algebra $H$, I wonder when the monoidal category of $H$-modules is left-closed, right-closed, and finally, under what circumstances right and left internal hom are isomorphic.
If I'm not mistaken, it seems that for $H$-modules $M,N$ one always has a natural $H$-action on $\text{Hom}(M,N)$ given by $(h.f)(m) := h^{\prime} f(S(h^{\prime\prime})m)$, and that provided with this $H$-action, evaluation and coevaluation $\text{Hom}(M,N)\otimes M\to N$ and $N\to\text{Hom}(M,N\otimes M)$ are $H$-linear. In particular $\text{Hom}(M,-)$ provides a right adjoint to $-\otimes M$ in $H\text{-Mod}$.
If the antipode $S$ of $H$ is invertible, one can put another $H$-action on $\text{Hom}(M,N)$, namely $(h.f)(m) := h^{\prime\prime} f(S^{-1}(h^{\prime})m)$, and this yields a right adjoint to $M\otimes -$ in $H\text{-Mod}$.
1) Where can I read about these things? I'm sure it is written down somewhere, but wasn't able to find a reference.
2) Under what circumstances are the two $H$-actions on $\text{Hom}(M,N)$ equal, or at least isomorphic? For example, they are equal if $H$ is cocommutative (in which case in particular $S=S^{-1}$). Is commutativity also sufficient? 
Thank you,
Hanno
 A: Let me try to answer to your question. As mentioned in this answer, if $(B,m,u,\Delta,\varepsilon)$ is a bialgebra over a field $\Bbbk$, then the category of left $B$-modules is always left and right closed.
Lemma: For every left $B$-module $V$ the functor ${_B\mathsf{Hom}}(B\otimes V,-)$ is right adjoint to the functor $-\otimes V:{_{B}\mathsf{Mod}}\to {_{B}\mathsf{Mod}}$. The left $B$-action on $B\otimes V$ is the diagonal one and the left $B$-action on ${_B\mathsf{Hom}}(B\otimes V,W)$ is given by
$$(b\cdot f)(a\otimes v) = f(ab\otimes v).$$
Moreover, the functor ${_B\mathsf{Hom}}(V\otimes B,-)$ is right adjoint to the functor $V\otimes -$. The left $B$-action on $V\otimes B$ is the diagonal one and the left $B$-action on ${_B\mathsf{Hom}}(V\otimes B,W)$ is given by
$$(b\cdot f)(v\otimes a) = f(v\otimes ab).$$
Proof: Define the assignments
$$\varphi:{_B\mathsf{Hom}}(U\otimes V,W)\to {_B\mathsf{Hom}}(U,{_B\mathsf{Hom}}(B\otimes V,W)), \\
\psi:{_B\mathsf{Hom}}(U,{_B\mathsf{Hom}}(B\otimes V,W))\to{_B\mathsf{Hom}}(U\otimes V,W) 
$$
given by
$$\varphi(f)(u):a\otimes v\mapsto f(a\cdot u\otimes v),\\
\psi(g):u\otimes v\mapsto g(u)(1\otimes v),$$
respectively, and the ones
$$\varphi':{_B\mathsf{Hom}}(V\otimes U,W)\to {_B\mathsf{Hom}}(U,{_B\mathsf{Hom}}(V\otimes B,W)), \\
\psi':{_B\mathsf{Hom}}(U,{_B\mathsf{Hom}}(V\otimes B,W))\to{_B\mathsf{Hom}}(V\otimes U,W) 
$$
given by
$$\varphi'(f)(u):v\otimes a\mapsto f(v\otimes a\cdot u),\\
\psi'(g):v\otimes u\mapsto g(u)(v\otimes 1),$$
respectively, for all $u\in U,v\in V,f\in {_B\mathsf{Hom}}(U\otimes V,W)$ and $g\in {_B\mathsf{Hom}}(U,{_B\mathsf{Hom}}(B\otimes V,W))$. Then check that these are inverses each other. $\square$
Proposition: For a Hopf algebra $H$ with antipode $S$ the functor ${\mathsf{Hom}}(V,-)$ is right adjoint to the functor $-\otimes V:{_{H}\mathsf{Mod}}\to {_{H}\mathsf{Mod}}$. The left $H$-action on ${\mathsf{Hom}}(V,W)$ is given by
$$(b\cdot f)(v) = \sum b_1f(S(b_2)v).$$
If, in addition, $S$ is invertible then ${\mathsf{Hom}}(V,-)$ is also right adjoint to the functor $V\otimes -$. The left $H$-action on ${\mathsf{Hom}}(V,W)$ is given by
$$(b\cdot f)(v) = \sum b_2f(S^{-1}(b_1)v).$$
Proof: Denote by a bullet $\bullet$ the given action on a module. For example, ${_\bullet V}\otimes {_\bullet W}$ denotes the diagonal left action on $V\otimes W$. Since $H$ is a Hopf algebra, the canonical map
$$\mathfrak{can}: {_\bullet H}\otimes V \to {_\bullet H}\otimes {_\bullet V}, \quad h\otimes v\mapsto \sum h_1\otimes h_2\cdot v$$
is an isomorphism of left $H$-modules with inverse
$$\mathfrak{can}^{-1}: {_\bullet H}\otimes {_\bullet V} \to {_\bullet H}\otimes V, \quad h\otimes v\mapsto \sum h_1\otimes S(h_2)\cdot v.$$
If you plug it into ${_H\mathsf{Hom}}(H\otimes V,W)$ you realise that
$${_H\mathsf{Hom}}({_\bullet H}\otimes {_\bullet V},{_\bullet W})\cong {_H\mathsf{Hom}}({_\bullet H}\otimes V,{_\bullet W})\cong {\mathsf{Hom}}(V,W)$$
(recall that for any algebra $A$, $A\otimes -:\mathsf{Vec}_{\Bbbk}\to {_A\mathsf{Mod}}$ is left adjoint of the forgetful functor). Now, if you induce via this isomorphism the left $H$-action on ${\mathsf{Hom}}(V,W)$ that comes from the left $H$-action on ${_H\mathsf{Hom}}({_\bullet H}\otimes {_\bullet V},{_\bullet W})$ of the previous lemma, you will find the one in the statement. Now, if in addition $S$ is invertible, then one can prove that also
$$\mathfrak{can}': V\otimes {_\bullet H} \to {_\bullet V}\otimes {_\bullet H}, \quad v\otimes h\mapsto \sum h_1\cdot v\otimes h_2$$
is an isomorphism of left $H$-modules with inverse
$$\left(\mathfrak{can}'\right)^{-1}: {_\bullet V}\otimes {_\bullet H} \to V \otimes {_\bullet H}, \quad v\otimes h\mapsto \sum S^{-1}\left(h_1\right)\cdot v\otimes h_2.$$
If you plug it into ${_H\mathsf{Hom}}(V\otimes H,W)$ you realize that
$${_H\mathsf{Hom}}({_\bullet V}\otimes {_\bullet H},{_\bullet W})\cong {_H\mathsf{Hom}}(V\otimes {_\bullet H},{_\bullet W})\cong {\mathsf{Hom}}(V,W)$$
and the action is the claimed one. $\square$
Concerning your last questions.

*

*I don't know where you may exactly read about this topic in this form. The argument above comes essentially from my PhD thesis. Maybe, if it is written somewhere, I would guess in B. Pareigis' or P. Schauenburg's works.

*Instead, concerning when the two are isomorphic, assume that they are. By uniqueness of the adjoint, this implies that the functors $-\otimes V$ and $V\otimes -$ are isomorphic for every $V$ and naturally in $V$ (this should be the counterpart of Schauenburg, Tannaka duality for arbitrary Hopf algebras, Remark 1.2.11). Therefore, modulo silly mistakes, the category of left modules becomes a braided monoidal category and so $H$ inherits a quasi-triangular structure (Kassel, Quantum Groups, Prop XIII.1.4). I would said (but I didn't check it right now) that the converse is true as well: if $H$ is quasi-triangular, then the category of modules should be left and right closed with forgetful functor to vector spaces preserving the internal homs (I mean, it should be true in general that a one-sided closed braided monoidal category is closed on both sides). Notice that, in view of Radford, On the antipode of a quasitriangular Hopf algebra, if $H$ is quasitriangular then the antipode is invertible.

I hope this answer your questions.
