If we have two Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ over a field $k$, and if we have a Lie algebra homomorphism $\mathfrak{g}\rightarrow \text{Der}\_k(\mathfrak{h})$, then we can define the semi-direct product $\mathfrak{g}\ltimes \mathfrak{h}$: as a $k$-linear space it is just $\mathfrak{g}\oplus\mathfrak{h}$ and the Lie bracket is given by
$$
[(g\_1,h\_1),(g\_2,h\_2)]=([g\_1,g\_2],[h\_1,h\_2]+g\_1\cdot h\_2-g\_2\cdot h\_1).
$$


Now we have the universal enveloping algebras $U(\mathfrak{g})$, $U(\mathfrak{h})$ and $U(\mathfrak{g}\ltimes \mathfrak{h})$. $\textbf{My question}$ is: could we form a semi-direct product $U(\mathfrak{g})\ltimes U(\mathfrak{h})$ such that
$$
U(\mathfrak{g})\ltimes U(\mathfrak{h})\cong U(\mathfrak{g}\ltimes \mathfrak{h})?
$$



Could we define the semi-direct product of two universal enveloping algebras?
I guess this is what is usually called the cross-product of Hopf algebras, restricted to the case of cocommutative Hopf algebras. The starting relevant paper should be this one, by Susan Montgomery:


<http://link.springer.com/chapter/10.1007/978-94-009-2985-2_22>


and of course also her book "Hopf algebras and their actions on rings", for the chapter in which she deals with crossed products (the statement you need should be the one at page 110).


Limiting oneself to the associative algebra part of the story is described in McConnell-Robson's book on Noncommutative Noetherian Rings at page 34: (crossed product of an associative algebra to a universal enveloping algebra, together with the case of the crossed product of two universal enveloping algebras).


(The definition of crossed product admits an incredible amount of generalzations, not always very easy to deal with, also because of overlapping terminolgies: smash product, bismash product etc...)


