nLab continuous homomorphisms of Lie groups are smooth

Contents

Context

Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

Contents

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.

References

Textbook accounts:

Lecture notes:

Last revised on September 3, 2021 at 11:16:31. See the history of this page for a list of all contributions to it.