Could we define the semi-direct product of two universal enveloping algebras?

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 $$U(\mathfrak{g})\ltimes U(\mathfrak{h})\cong U(\mathfrak{g}\ltimes \mathfrak{h})?$$

• In your last line you might just define the left side to equal the right side. Obviously this isn't helpful, but what kind of intrinsic properties would you expect a "semi-direct product" of such Hopf algebras to have? Sep 19 '13 at 19:53
• The definition should extend the special case of the direct product, i.e., $U(\mathfrak{g}\times \mathfrak{h})\simeq U(\mathfrak{h})\otimes U(\mathfrak{g})$. Sep 19 '13 at 20:49
• @JimHumphreys Yes, you're right. However, I'm still not clear how to define semi-direct product through universal properties, even the semi-direct product of two groups. The only way I know is to construct it by hand. Maybe that's the reason that why I cannot define the semi-direct product of two universal enveloping algebras. Sep 19 '13 at 21:31
• @DietrichBurde Sure! We should expect that the semi-direct product extends the tensor product. But this also leads to a problem: in the category of $k$-algebras, the tensor product does not have good universal properties, as far as I know. For example, $A\otimes B$ is not the coproduct of $A$ and $B$ if we do not restrict to commutative algebras. Sep 19 '13 at 21:36
• $A \otimes B$ is the "commutative coproduct": it's the universal thing with a map from $A$ and $B$ whose images commute. Sep 19 '13 at 22:34

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: