This may be totally trivial or wrong. I am just posting this because I am sick and tired of trying to understand this myself and I am sure someone out here can just answer it out of his head in 2 minutes.


Let $G$ be a finite group, and $k$ an algebraically closed field whose characteristic is not a divisor of $\left|G\right|$. The Maschke theorem in its standard form states that 
$$
\displaystyle k\left[G\right]\cong\bigoplus\_{V\text{ irreducible }k\left[G\right]\text{-module}}\mathrm{End}V
$$ 
as $k$-algebras. This quickly yields that 
$$
\displaystyle k\left[G\right]\cong\bigoplus\_{V\text{ irreducible }k\left[G\right]\text{-module}}\mathrm{End}V
$$ 
as left $k\left[G\right]$-modules, where the $k\left[G\right]$-module structure on $\mathrm{End}V$ is given by $\left(gF\right)\left(v\right)=gF\left(v\right)$ for every $g\in k\left[G\right]$, $f\in\mathrm{End}V$ and $v\in V$.


Now, I've overheard that there exists a stronger form of this, stating that
$$
\displaystyle k\left[G\right]\cong\bigoplus\_{V\text{ irreducible }k\left[G\right]\text{-module}}\mathrm{End}V
$$ 
as left $k\left[G\times G\right]$-modules, where the $k\left[G\times G\right]$-module structure on $k\left[G\right]$ is defined by $\left(g,h\right)u=guh^{-1}$ for any $g\in G$, $h\in G$ and $u\in k\left[G\right]$, and the $k\left[G\times G\right]$-module structure on $\mathrm{End}V$ is defined by $\left(g,h\right)F=gFh^{-1}$ for any $g\in G$, $h\in G$ and $F\in \mathrm{End}V$. This would follow from 
$$
\displaystyle k\left[G\right]\cong\bigoplus\_{V\text{ irreducible }k\left[G\right]\text{-module}}\mathrm{End}V
$$ 
as $k$-Hopf algebras, where the Hopf algebra structure on $k\left[G\right]$ is the standard one ($S\left(g\right)=g^{-1}$ for every $g\in G$), and the Hopf algebra structure on $\mathrm{End}V$ is the standard one as well.


[**EDIT:** as the comments explained, there is no such thing as a standard Hopf algebra structure on $\mathrm{End}V$, and so "$k$-Hopf algebras" should be "$k$-algebras".]


Is this correct, and how can this be proven?



Apocryphal Maschke theorem?
The result about bimodules is true, and standard. Here is one way to see it.


By Frobenius reciprocity, $\operatorname{Hom}\_G(V,k[G]) = \operatorname{Hom}\_k(V,k)$, since $k[G]$ is the induction (or coinduction, depending on your terminology) of the trivial representation of the trivial subgroup of $G$ to $G$.


Since Frobenius reciprocity is functorial, one easily sees that this canonical isomorphism
is an isomorphism of right $G$-representations, where the source has a right $G$-action coming from the right $G$-action on $k[G]$, and the target has a right $G$-action coming as the transpose of the left $G$-action on $V$.


Now by Maschke's semisimplicity theorem, we know that 
$k[G] = \bigoplus\_V V \otimes\_k \operatorname{Hom}\_G(V,k[G])$, where the sum is over all irreducible left $G$-representations. (Indeed, Maschke shows that this is true
for any left $G$-module in place of $k[G]$.) Again, this is a natural isomorphism, and so respects the right $G$-actions on source and target.


Combined with the preceding computation, we find that
$k[G] = \bigoplus\_V V\otimes\_k \operatorname{Hom}\_k(V,k) = \bigoplus\_V \operatorname{End}(V),$ as both left and right $G$-modules,
as required.


[Edit:] Leonid's remark about $k$ being needing to be big enough in his answer below is correct. Each simple $V$ comes equipped with an associated division algebra of $G$-endomorphisms
$A\_V := \operatorname{End}\_G(V)$. The representation $V$ is absolutely irreducible (i.e stays irred. after passing to any extension field) if and only if $A\_V = k$. When we consider $\operatorname{Hom}\_k(V,W)$ for another left $G$-module $W$, this is naturally an $A\_V$-module, and Maschke's theorem
will say that $W = \bigoplus\_V \operatorname{Hom}\_k(V,W)\otimes\_{A\_V} V$. (I have written the factors in the tensor product in this order because $V$ is naturally a left $A\_V$-module (if we think of endomorphisms acting on the left), and then $\operatorname{Hom}\_k(V,W)$ becomes a right $A\_V$-module.)


So in the case of $W$ being the group algebra, we have
$$k[G] = \bigoplus\_V \operatorname{Hom}\_k(V,k)\otimes\_{A\_V} V$$
(an isomorphism of $G$-bimodules).


If all the $V$ are absolutely irreducible, e.g. if $k$ is algebraically closed,
then all the $A\_V$ just equal $k$, and the preceding direct sum reduces to what I wrote above, and what was written in the question.


