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continuous homomorphisms of Lie groups are smooth

Contents

Context

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Statement

Proposition

Let G,HGrp(SmthMfd)Grp(undrl)Grp(TopSp)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 GHG \xrightarrow{\;} H is smooth.

In other words, the function of hom-sets

Grp(SmthMfd)(G,H)undrl 1Grp(TopSp)(G,H) 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)

References

Textbook accounts:

Lecture notes:

Last revised on September 3, 2021 at 07:16:31. See the history of this page for a list of all contributions to it.