Regular Lie groups and a theorem of Lie-Palais

TL;DR

This paper simplifies the proof of the Lie-Palais theorem for effective Lie group actions on compact manifolds, using regular Lie groups modeled on locally convex spaces, and addresses a related problem from 1972.

Contribution

It provides an elementary proof of the Lie-Palais theorem in the effective case, removing the need for the Lie-Cartan theorem, and offers a partial solution to a 1972 problem.

Findings

Elementary proof of Lie-Palais theorem for effective actions

Elimination of Lie-Cartan theorem in the proof

Partial solution to a 1972 problem

Abstract

In 1984 Milnor had shown how to deduce the Lie-Palais theorem on integration of infinitesimal actions of finite-dimensional Lie algebras on compact manifolds from general theory of regular Lie groups modelled on locally convex spaces. We show how, in the case of effective action, one can eliminate from Milnor's argument the abstract Lie-Cartan theorem, making the deduction rather elementary. A machinery employed in the proof provides a partial solution to a problem examined in 1972 by van Est and \'Swierczkowski.