# Are group theoretic splittings of Lie groups automatically differentiable?

Suppose that $G$ is a Lie group, and that $N$ is a normal Lie subgroup of $G$. Then $G / N$ is also a Lie group.

If $0 \to N \to G \to G/N \to 0$ splits as groups (i.e. $G$ is a semidirect product of $N$ and $G/N$ as abstract groups, i.e. there is some not necessarily smooth section $G/N$ to $G$), with splitting $\gamma : G / N \to G$, then does $G$ also split as Lie groups, i.e. the image of $\gamma$ is a Lie subgroup of $G$?

A priori I see no reason for $\gamma$ to be smooth, also there is the issue that the smooth image of a Lie group may not be a Lie group.

If the image of the splitting $\gamma$ was a manifold, then the restriction $p : G \to G / N$ of the projection to it is smooth bijective homomorphism between Lie groups, hence a diffeomorphism. So in this case $\gamma$ would be smooth.

So the question is really: Is $\gamma(G/N)$ a manifold? (I suspect no, but is there a good example?)

• @JasonDeVito I am assuming that $N$ is a Lie subgroup (i.e. is a closed submanifold of some open set). So it is automatically closed. I have corrected this typo, sorry for the confusion. Feb 23, 2016 at 2:59
• Some (most?) people call the irrational line a Lie subgroup. Feb 23, 2016 at 3:04
• @Timkinsella Not the text I am following, but point taken. Feb 23, 2016 at 3:06

Suppose that $N$ is commutative, the splitting extensions are classified by $H^1(G,N)$ which represents here the discrete or the continuous cohomology. If $N$ is $Q$ and $R$ the discrete cohomology and the Lie cohomology do not coincide generally. You can see for example this paper of Milnor, where the (co) homology relatively to different coefficients are discussed.
• I think the OP has in mind a closed Lie subgroup $N$. Feb 23, 2016 at 5:01