nLab continuous homomorphisms of Lie groups are smooth

Statement

Proposition

Let $G, H \,\in\, Grp(SmthMfd) \xrightarrow{\;Grp(undrl)\;} Grp(TopSp)$ be Lie groups (finite dimensional) with underlying topological groups denoted by the same symbol.

Then every continuous group homomorphism $G \xrightarrow{\;} H$ is smooth.

In other words, the function of hom-sets

$$Grp(SmthMfd) (G,\,H) \xrightarrow{undrl_1} Grp(TopSp) (G,\,H)$$

is a bijection.

This follows by applying Cartan's closed subgroup theorem to the graph of the homomorphism.

(e.g. Wang 2013, Cor. 1.7, Hall 2015, Cor. 3.50)

Related concepts

• Cartan's closed subgroup theorem

• smooth Oka principle

References

Textbook accounts:

• Brian C. Hall, Cor. 3.50 in: Lie Groups, Lie Algebras, and Representations, Springer 2015 (doi:10.1007/978-3-319-13467-3)

Lecture notes:

• Zuoqin Wang, Cor. 1.7 in: Cartan’s closed subgroup theorem, Lecture 11 in: 李群引论, 2013 (pdf, pdf)