Questions about $\mathrm{Hom}_k(V, W) \cong W \otimes V^*$.

Question

I am reading A Guide to Quantum Groups by Chari and Pressley. I have some questions on page 111 about $\mathrm{Hom}_k(V, W) \cong W \otimes V^*$.

If $V$ and $W$ are representations of $A$, we make $\mathrm{Hom}_k(V, W)$ into a representation of $A$ by setting $$ \newcommand{\act}{\mathbin{.}} (a \act f)(v) = \sum_i a_i \act f(S(a^i) \act v) \,, $$ for $f ∈ \mathrm{Hom}_k(V, W)$, $v ∈ V$, where $Δ(a) = \sum_i a_i \otimes a^i$. If $W$ is the trivial representation, this is of course the dual $V^*$. We leave it to the reader to check that, if $k$ is a field (say), the canonical vector space isomorphism $\mathrm{Hom}_k(V, W) \cong W \otimes V^*$ is actually an isomorphism of representations of $A$ (but this is not true for the vector space isomorphism $\mathrm{Hom}(V, W) \cong V^* \otimes W$!).

(Original image)

Let $W, V$ be $A$-modules. It is said that the vector space isomorphism $\varphi \colon \mathrm{Hom}_k(V, W) \cong W \otimes V^*$ commutes with the $A$-action. But the vector space isomorphism $\psi \colon \mathrm{Hom}_k(V, W) \cong V^* \otimes W$ does not commute with the $A$-action. Why the vector space isomorphism $\psi \colon \mathrm{Hom}_k(V, W) \cong V^* \otimes W$ does not commute with the $A$-action? Thank you very much.

Answer 1

It should be remarked that this question is asked in the context of Hopf algebras over the field $k$.

You seem to accept that the claim of the book is true, so $\operatorname{Hom}_k(V,W)\cong W\otimes V^*$ is an isomorphism not only as $k$-modules but also as $A$-modules.

Imagine both claims hold, then $W\otimes V^*\cong V^*\otimes W$. Note that the underlying vector space isomorphism here is simply the flip map $$v\otimes w\mapsto w\otimes v.$$ Further note that each finite-dimensional $A$-module $X$ actually arises as $V^*$ for some $A$-modules $V$. Hence we might as well ask if $W\otimes X\cong X\otimes W$ as $A$-modules. But the tensor product action is given by the coproduct $\Delta$, hence this implies that $$\sum_i a_i\cdot w\otimes a^i\cdot v=\sum_i a^i\cdot w\otimes a_i\cdot v .$$ But it is easy to find Hopf algebras which are not cocommutative. For a concrete counterexample, take e.g. the Hopf algebra of functions on a non-abelian finite group (or $U_q(\mathfrak{sl_2})$ if you like). All of these Hopf algebras $A$ have a regular representation, and one sees that the claimed identity is false when applied to the tensor product $A\otimes A$ and evaluated at the element $1\otimes 1$. It is however true for cocommutative Hopf algebras.

Rmk: Requiring that not this specific isomorphism $W\otimes X\cong X\otimes W$ to be an isomorphism of $A$-modules, but instead an arbitrary (natural) system of $k$-vs isomorphisms to exist, which are $A$-linear leads to the notion of quasi-triangular Hopf algebras.