Book contents
- Frontmatter
- Contents
- Preface
- Chapter 1 Analytic manifolds
- Chapter 2 Lie groups and Lie algebras
- Chapter 3 The Campbell-Baker-Hausdorff formula
- Chapter 4 The geometry of Lie groups
- Chapter 5 Lie subgroups and subalgebras
- Chapter 6 Characterisations and structure of compact Lie groups
- Appendix A Abstract harmonic analysis
- Biblography
- Index
Chapter 6 - Characterisations and structure of compact Lie groups
Published online by Cambridge University Press: 02 December 2009
- Frontmatter
- Contents
- Preface
- Chapter 1 Analytic manifolds
- Chapter 2 Lie groups and Lie algebras
- Chapter 3 The Campbell-Baker-Hausdorff formula
- Chapter 4 The geometry of Lie groups
- Chapter 5 Lie subgroups and subalgebras
- Chapter 6 Characterisations and structure of compact Lie groups
- Appendix A Abstract harmonic analysis
- Biblography
- Index
Summary
The first section of this chapter is concerned with establishing a number of conditions which are equivalent to a compact group being Lie. These include the conditions that G has no small subgroups and that G is isomorphic to a closed subgroup of some unitary group. Thus all compact groups are linear; section 2 contains an example due to Birkhoff of a connected Lie group which is not linear. Before providing the structure theorems of section 4 it is necessary (in section 3) to analyse simple and semisimple Lie algebras and Lie groups. In the main this will be done using a powerful tool, the Killing form of a Lie algebra. Such an analysis allows a proof of the fact that all semisimple connected Lie subgroups of a compact Lie group are closed. This result finally completes the mise en scène for the entrance of the promised structure theorems; section 4 contains these theorems for compact connected Lie groups while section 5 contains them for arbitrary compact connected groups.
Compact groups and Lie groups
The main result in this section is a list of necessary and sufficient conditions for a compact group to be a Lie group.
- Type
- Chapter
- Information
- Lie Groups and Compact Groups , pp. 115 - 159Publisher: Cambridge University PressPrint publication year: 1977