Introduction to Lie groups, isometric and adjoint actions and some generalizations

TL;DR

This paper provides an introductory overview of Lie groups, Lie algebras, and their actions, connecting classical theories to recent research on singular Riemannian foliations and generalizations.

Contribution

It offers a concise introduction to Lie theory with new insights into isoparametric submanifolds, polar actions, and singular Riemannian foliations, including recent research developments.

Findings

Connections between Lie groups and isoparametric submanifolds

Survey of recent results on singular Riemannian foliations

New examples of foliations via surgery and suspension

Abstract

The main purpose of these lecture notes is to provide a concise introduction to Lie groups, Lie algebras, and isometric and adjoint actions, aiming mostly at advanced undergraduate and graduate students. In addition, the connection between such classic theories and the research area of the first author is explored. Namely, generalizations to isoparametric submanifolds, polar actions and singular Riemannian foliations with sections (s.r.f.s.) are mentioned. The first chapters cover basic concepts, giving results on adjoint representation, closed subgroups, bi-invariant metrics, Killing forms and splitting in simple ideals. In the following chapters, proper and isometric actions are recalled together with adjoint action and foliations, mostly concerning the Weyl group, normal slices and Dynkin diagrams. A special focus is given to maximal tori and roots of compact Lie groups, exploring its connection with isoparametric submanifolds and polar actions. Furthermore, in the last chapter, a survey on recent research results on s.r.f.s. is given. In this revised version, more details about fiber bundles, proper and isometric actions are explored, and further exercises and examples were added. It also features new sections with examples of singular Riemannian foliations constructed with surgery and suspension of homomorphisms. This is still a preliminary version and we expect to improve it in the future. We would be grateful for any kind of suggestions.