Actions of Lie groups and Lie algebras on manifolds

TL;DR

This paper explores the actions of Lie groups and Lie algebras on manifolds, examining effectiveness, smoothness, fixed points, and perturbations, and presents new results on the existence of effective smooth but not analytic actions.

Contribution

It introduces new examples of solvable Lie algebras that act smoothly but not analytically on all n-manifolds, expanding understanding of Lie group actions.

Findings

Existence of solvable Lie algebras with effective smooth actions on all n-manifolds.

Some Lie algebras lack effective analytic actions on certain or all n-manifolds.

Summary of classical results on Lie group actions on manifolds.

Abstract

Questions of the following sort are addressed: Does a given Lie group or Lie algebra act effectively on a given manifold? How smooth can such actions be? What fxed-point sets are possible? What happens under perturbations? Old results are summarized, and new ones presented, including: For every integer n there are solvable (in some cases, nilpotent) Lie algebras g that have effective C-infinity actions on all n-manifolds, but on some (in many cases, all) n-manifolds, g does not have effective analytic actions