Let $H$ be a bialgebra and $A$ a $H$-module algebra. The cross product $A \rtimes H$ is defined as follows. As a vector space $A \rtimes H = A \otimes H$. The multiplication on $A \rtimes H$ is defined as follows:
\begin{align}
(a \otimes h)(b \otimes g) = \sum a(h\_{(1)}.b) \otimes h\_{(2)}g,
\end{align}
where $a, b \in A$, $h, g \in H$, $\Delta(h) = \sum h\_{(1)} \otimes h\_{(2)}$.


Is there a comultiplication $\Delta$ on $A \rtimes H$ such that the cross product $A \rtimes H$ is a bialgebra? If $\Delta$ does not always exist, under what conditions there is a comultiplication $\Delta$ on $A \rtimes H$ such that $A \rtimes H$ is a bialgebra? Thank you very much.


Edit: we add one more condition: suppose that $A$ is a bialgebra. In this case, is there a comultiplication $\Delta$ on $A \rtimes H$ such that the cross product $A \rtimes H$ is a bialgebra? If $\Delta$ does not always exist, under what conditions there is a comultiplication $\Delta$ on $A \rtimes H$ such that $A \rtimes H$ is a bialgebra? Thank you very much.



Is the cross product $A \rtimes H$ a bialgebra?
In the case that $A$ is a braided Hopf algebra in $\_H^H YD$ you can form $A \# H$ with the multiplication you already wrote down. The coproduct uses the fact that this kind of $A$ also has a left $H$ coaction as well given by $\delta r^{(2)} = r^{(2)}\_{(-1)} \otimes r^{(2)}\_{(0)}$


$$
\Delta ( r \# h) = ( r^{(1)} \# r^{(2)}\_{(-1)} h\_{(1)} ) \otimes ( r^{(2)}\_{(0)} \# h\_{(2)})
$$


If you want finite dimensional $H$, then you can say in terms of modules for $D(H)$.


Chapters 6 and 7 of Foundations of Quantum Group Theory discusses extension problems such as these. See there for compatibility conditions too. You might also look in <http://www.sciencedirect.com/science/article/pii/S0021869384710118>


