Let $H$ be a bialgebra and $B$ an $H$-module. The cross product $B \rtimes H$ of $B$ and $H$ is $B \otimes H$ as a vector space and the multiplication in $B \rtimes H$ is defined as follows: $(a \otimes g)(b \otimes h) = a(g\_{(1)}.b) \otimes (g\_{(2)}h)$, where $a,b \in B$, $g, h \in H$, $\Delta(g) = g\_{(1)} \otimes g\_{(2)}$. What are the motivations of the definition of cross product? Thank you very much.



Motivations of cross product
It's the Hopf-algebraic version of the semidirect product of groups. To see this, just consider the case you have groups $G, H, G\rtimes H$. The group rings (over, say, a field $k$) are Hopf algebras (with $\Delta(g)=g\otimes g$); compare $kG\rtimes kH$ and $k(G\rtimes H)$ and you will see they have the same multiplication.


