Book contents
- Frontmatter
- Contents
- Foreword
- Lie Algebras and Root Systems
- Lie Groups
- Introduction
- 1 Examples
- 2 SU2, SO3, and SL2ℝ
- 3 Homogeneous spaces
- 4 Some theorems about matrices
- 5 Lie theory
- 6 Representation theory
- 7 Compact groups and integration
- 8 Maximal compact subgroups
- 9 The Peter-Weyl theorem
- 10 Functions on ℝn and Sn-1
- 11 Induced representations
- 12 The complexification of a compact group
- 13 The unitary and symmetric groups
- 14 The Borel-Weil theorem
- 15 Representations of non-compact groups
- 16 Representations of SL2ℝ
- 17 The Heisenberg group
- Linear Algebraic Groups
- Notes and references
- Bibliography
- Index
Introduction
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Foreword
- Lie Algebras and Root Systems
- Lie Groups
- Introduction
- 1 Examples
- 2 SU2, SO3, and SL2ℝ
- 3 Homogeneous spaces
- 4 Some theorems about matrices
- 5 Lie theory
- 6 Representation theory
- 7 Compact groups and integration
- 8 Maximal compact subgroups
- 9 The Peter-Weyl theorem
- 10 Functions on ℝn and Sn-1
- 11 Induced representations
- 12 The complexification of a compact group
- 13 The unitary and symmetric groups
- 14 The Borel-Weil theorem
- 15 Representations of non-compact groups
- 16 Representations of SL2ℝ
- 17 The Heisenberg group
- Linear Algebraic Groups
- Notes and references
- Bibliography
- Index
Summary
These notes are an expanded version of the seven hours of lectures I gave at Lancaster. I have kept to the original plan and policy, which perhaps need some explanation. Roughly speaking, the contents are what I should like my own graduate students to know about Lie groups, and my general idea was to show how the theory is a natural continuation of basic linear algebra. As root systems and the classification of semisimple Lie algebras were treated in the companion lecture courses I felt I had an excuse for concentrating firmly on the general linear groups. But in any case I believe that is the right way to approach the subject: the taxonomic side of the theory is not to my taste.
I tried to make my lectures useful to people with rather different amounts of mathematical knowledge and sophistication. That means the level is uneven: remarks aimed at the more advanced readers are scattered throughout, and are meant to be ignored by others. I hope the chapters can be read in almost any order: I tried to make them fairly independent. The first four are devoted to a survey of concrete examples of the theory to be developed. This is mainly “undergraduate” material, and so I put it before the formal definition of a Lie group in Chapter 5. But it does not need to be read in advance, and sometimes it uses terminology which is defined only later.
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- Information
- Lectures on Lie Groups and Lie Algebras , pp. 47 - 48Publisher: Cambridge University PressPrint publication year: 1995