Given two cosemisimple Hopf algebras $H,G$ over ${\mathbb C}$, denote their usual (not braided) tensor product by $G \otimes H$. What conditions do we need to impose on the Hopf algebras to ensure that the corepresentations of $G \otimes H$ are direct sums of corepresentations of the form $V \otimes W$, where $V$ is a corepresentation of $G$ and $W$ is a corepresentation of $H$? 


I have heard a rumour that this is true when $G$ and $H$ can be completed to compact quantum groups. However, I can't find a reference, and I believe that it should be true for a larger class of Hopf algebras.



Corepresentations of Tensor Products of Hopf Algebras
Let $A$ be a coalgebra over a field $\mathbb K$ and $\mathcal A = \mathrm{Comod}^A$ its category of comodules. A well-known result of Sweedler writes $A = \mathrm{colim}\_i A\_i$ where $A\_i$ are finite-dimensional subcoalgebras of $A$. Let $A\_i^\*$ denote the linear dual to $A\_i$ and $\mathcal A\_i = \mathrm{Mod}\_{A\_i^\*} = \mathrm{Comod}^{A\_i}$ the category of $A\_i^\*$-modules, equivalently the category of $A\_i$-comodules. As observed in [my paper with Brandenburg and Chirvasitu](http://www.tac.mta.ca/tac/volumes/30/23/30-23abs.html), $\mathcal A = \mathrm{colim} \mathcal A\_i$ is the colimit (over the same diagram) in the bicategory of $\mathbb K$-linear locally presentable categories. Here and throughout I sloppily write "$=$" for "there is a canonical equivalence".


Tensor products of coalgebras distribute over colimits of coalgebras, and tensor products of categories distribute over colimits of categories. Moreover, it is well-known that if $R$ and $S$ are $\mathbb K$-algebras, then $\mathrm{Mod}\_{R \otimes S} = \mathrm{Mod}\_R\boxtimes \mathrm{Mod}\_S$. (This can be checked directly from the universal property of the tensor product.)


Suppose that $B = \mathrm{colim}\_j B\_j$ is another coalgebra over $\mathbb K$ with $B\_j$ its finite-dimensional subcoalgebras, and $\mathcal B = \mathrm{Comod}^B$ and $\mathcal B\_j = \mathrm{Mod}\_{B\_j^\*}$ as above. We have
$$ \begin{aligned}\mathcal A \boxtimes \mathcal B & = (\mathrm{colim}\_{i} \mathcal A\_i) \boxtimes (\mathrm{colim}\_j \mathcal B\_j) \\ & = \mathrm{colim}\_{i,j} (\mathcal A\_i \boxtimes \mathcal B\_j) \\ & = \mathrm{colim}\_{i,j} (\mathrm{Mod}\_{A\_i}^\* \boxtimes \mathrm{Mod}\_{B\_j^\*}) \\ & = \mathrm{colim}\_{i,j} (\mathrm{Mod}\_{A\_i^\* \otimes B\_j^\*}) \\ & = \mathrm{colim}\_{i,j} (\mathrm{Comod}^{A\_i \otimes B\_j}) \\ & = \mathrm{Comod}^{\mathrm{colim}\_{i,j} A\_i \otimes B\_j} \\ & = \mathrm{Comod}^{A\otimes B}. \end{aligned}$$


On the other hand, the tensor product of categories can be presented as follows. For arbitrary $\mathbb K$-linear locally presentable categories $\mathcal A$, $\mathcal B$, objects of $\mathcal A \boxtimes \mathcal B$ are colimits over objects of the form $V \boxtimes W$ subject to the relation that $\mathrm{colim}\_m (V\_m \boxtimes W) = (\mathrm{colim}\_m V\_m) \boxtimes W$, and similarly in the $W$-variable, where on the LHS the colimit is computed in $\mathcal A \boxtimes \mathcal B$ and on the RHS the colimit is computed in $\mathcal A$.


It follows from this presentation that if $\mathcal A$ and $\mathcal B$ are both semisimple categories, then so is $\mathcal A\boxtimes \mathcal B$. Indeed, suppose that $\mathcal A$ is semisimple and let $\mathcal I = \{I\_1,\dots\}$ be a complete set of simples. By "semisimple" I mean that every object of $\mathcal A$ is a direct sum of objects from $\mathcal I$. Let $\mathbb K\_m = \mathrm{End}(I\_m)$; by Schur's lemma, it is a division ring. Then $\mathcal A = \bigoplus\_m \mathrm{Mod}\_{\mathbb K\_m}$. If $\mathcal B$ is also semisimple with simples $\{J\_1,\dots\}$ and $\mathrm{End}(J\_n) = \mathbb L\_n$, then
$$ \mathcal A \boxtimes \mathcal B = \bigoplus\_{m,n} \mathrm{Mod}\_{\mathbb K\_m} \boxtimes \mathrm{Mod}\_{\mathbb L\_n} = \bigoplus\_{m,n} \mathrm{Mod}\_{\mathbb K\_m \boxtimes \mathbb L\_n}.$$


Suppose now that $\mathbb K$ is algebraically closed and $\mathcal A = \mathrm{Comod}^A$ and $\mathcal B = \mathrm{Comod}^B$. Then $\mathbb K\_m = \mathbb L\_n = \mathbb K$ for all $m,n$. It follows that every object of $\mathcal A \boxtimes \mathcal B$ is a direct sum of objects of the form $I\_m \boxtimes J\_n$, as you wanted. (Under the equivalence $\mathcal A \boxtimes \mathcal B = \mathrm{Comod}^{A\otimes B}$, $I \boxtimes J$ corresponds to the $A\otimes B$-comodule $I \otimes J$.)


If $\mathbb K$ is not algebraically closed, then one can find many examples where $\mathbb K\_m \boxtimes \mathbb L\_n$ is not a division ring. It is always, however, a semisimple algebra, and so its module theory is semisimple with simples given by direct summands of the regular module. Thus in this case every object of $\mathrm{Comod}^{A\otimes B}$ is a direct sum *of direct summands* of objects of the form $I\_m \boxtimes J\_n$.


So as you can see, the answer to your question correctly interpreted is "yes", and it has nothing to do with Hopf algebras or quantum groups.


