A certain decomposition of a semisimple Hopf algebra $\newcommand{\Irr}{\mathrm{Irr}}$
Let $H$ be a finite-dimensional semisimple Hopf algebra over an algebraically closed field $k$, and let $\Irr(H)$ be the set of (choices of representatives of) isomorphism classes of simple $H$-modules.
I was told that
\begin{align}\label{eq:question}
    H \cong \bigoplus_{U\in\Irr(H)} U\otimes U^*
\end{align}
is a consequence of Artin-Wedderburn, but I am not well-read in basic facts, I think.
Artin-Wedderburn tells me that
\begin{align}
    H \cong \bigoplus_{U\in\Irr(H)} k^{m_U\times m_U},
\end{align}
where $m_U$ is the multiplicity of $U$ in $H$.
The right hand side of the equation in question is the same as
\begin{align}
    \bigoplus_{U\in\Irr(H)} k^{\dim U\times \dim U}.
\end{align}
Why are the last two expressions now the same?
 A: Note that Artin-Wedderburn says that 
$$ H \to \bigoplus_{U \in \text{Irr}(H)}  \text{End}_k(U), h \mapsto (u \mapsto h u)_{U}$$
is an isomorphism of $k$-algebras. This is either what you show when proving Artin-Wedderburn or it follows from the isomorphism $H \cong \bigoplus_{U} k^{m_U \times m_U}$ as it clearly holds for the algebra on the right.
Also note that we have an isomorphism 
$$ U \otimes U^* \to \text{End}_k(U), v \otimes f \mapsto (u \mapsto f(u)v) $$
of $k$-vector spaces.
This gives $$ H \cong \bigoplus_{U} U \otimes U^* $$ as $k$-vector spaces.
Note that $\text{End}_k(U)$ becomes a $H$-module via $(h\varphi)(u) = h\varphi(u)$
for $h \in H$, $u \in U$ and $\varphi \in \text{End}_k(U)$ and $U \otimes U^*$
is an $H$-module via $h (u \otimes f)) = (h u) \otimes f$.
Given these module structures, the above isomorphisms actually become isomorphisms of $H$-modules and so $$H \cong \bigoplus_{U} U \otimes U^*$$ as $H$-modules given the above module structure of $U \otimes U^*$.
However, there is a huge danger here and that is that these module structures are not the ones we would expect in the context of Hopf-algebras.
The endomorphism ring $\text{End}_{k}(U)$ usually becomes an $H$-module via $$(h\varphi)(u) = \sum h_1 \varphi(S(h_2)u)$$ for $h \in H$, $u \in U$ and $\varphi \in \text{End}_k(U)$ and $U \otimes U^*$ becomes an $H$-module via $$h(u \otimes f) = \sum (h_1u) \otimes (h_2f)$$
Note that with these module structures, the isomorphism
$$ U \otimes U^* \to \text{End}_k(U), v \otimes f \mapsto (u \mapsto f(u)v) $$
is still an isomorphism of $H$-modules, however,
$$H \to \text{End}_k(U), h \mapsto (u \mapsto h u) $$
is not an $H$-module homomorphism in this case.
In fact, we do not have $H \cong \bigoplus_{U} U \otimes U^*$ for a semisimple Hopf algebra $H$ with this second $H$-module structure unless $H = k$. In fact, the map
$$U \otimes U^* \to k, u \otimes f \mapsto f(u)$$ is an $H$-module homomorphism to the trivial $H$-module. If we had $H \cong \bigoplus_{U} U \otimes U^*$ then the trivial module would have multiplicity at least $|\text{Irr}(U)|$ in $H$. By Artin-Wedderburn, the multiplicity of the trivial module in $H$ is its dimension and so we have to have $|\text{Irr}(U)| = 1$ and thus $H = k$.
