This textbook provides an essential introduction to Lie groups, presenting the theory from its fundamental principles. Lie groups are a special class of groups that are studied using differential and integral calculus methods. As a mathematical structure, a Lie group combines the algebraic group structure and the differentiable variety structure. Studies of such groups began around 1870 as groups of symmetries of differential equations and the various geometries that had emerged. Since that time, there have been major advances in Lie theory, with ramifications for diverse areas of mathematics and its applications.
Each chapter of the book begins with a general, straightforward introduction to the concepts covered; then the formal definitions are presented; and end-of-chapter exercises help to check and reinforce comprehension. Graduate and advanced undergraduate students alike will find in this book a solid yet approachable guide that will help them continue their studies with confidence.
Author(s): Luiz Antonio Barrera San Martin
Series: Latin American Mathematics Series
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2021
Language: English
Pages: 371
City: Cham
Tags: Lie Groups, Haar Measure, Representations, Lie Algebras, Group Actions
Preface
Acknowledgement
Contents
1 Introduction
1.1 Exercises
Part I Topological Groups
Overview
2 Topological Groups
2.1 Introduction
2.2 Neighborhoods of Identity
2.3 Metrizable Groups
2.4 Homomorphisms
2.5 Subgroups
2.6 Group Actions
2.6.1 Algebraic Description
2.6.2 Continuous Actions
2.7 Quotient Spaces
2.7.1 Quotient Groups
2.7.2 Compact and Connected Groups
2.8 Homeomorphism G/Gx→G·x
2.9 Examples
2.10 Exercises
3 Haar Measure
3.1 Introduction
3.2 Construction of Haar Measure
3.3 Uniqueness
3.4 Modular Function
3.5 Exercises
4 Representations of Compact Groups
4.1 Representations
4.2 Schur Orthogonality Relations
4.3 Regular Representations
4.4 Peter–Weyl Theorem
4.5 Exercises
Part II Lie Groups and Algebras
Overview
5 Lie Groups and Lie Algebras
5.1 Definition
5.2 Lie Algebra of a Lie Group
5.2.1 Invariant Vector Fields
5.3 Exponential Map
5.4 Homomorphisms
5.4.1 Representations
5.4.2 Adjoint Representations
5.5 Ordinary Differential Equations
5.6 Haar Measure
5.7 Exercises
6 Lie Subgroups
6.1 Definition and Examples
6.2 Lie Subalgebras and Lie Subgroups
6.3 Ideals and Normal Subgroups
6.4 Limits of Products of Exponentials
6.5 Closed Subgroups
6.6 Path Connected Subgroups
6.7 Manifold Structure on G/H, H Closed
6.8 Exercises
7 Homomorphisms and Coverings
7.1 Homomorphisms
7.1.1 Immersions and Submersions
7.1.2 Graphs and Differentiability
7.2 Extensions of Homomorphisms
7.3 Universal Covering
7.4 Appendix: Covering Spaces (Overview)
7.5 Exercises
8 Series Expansions
8.1 The Differential of the Exponential Map
8.2 The Baker–Campbell–Hausdorff Series
8.3 Analytic-Manifold Structure
8.4 Exercises
Part III Lie Algebras and Simply Connected Groups
Overview
9 The Affine Group and Semi-Direct Products
9.1 Automorphisms of Lie Groups
9.2 The Affine Group
9.3 Semi-Direct Products
9.4 Derived Groups and Lower Central Series
9.5 Exercises
10 Solvable and Nilpotent Groups
10.1 Solvable Groups
10.2 Nilpotent Groups
10.3 Exercises
11 Compact Groups
11.1 Compact Lie Algebras
11.2 Finite Fundamental Group
11.2.1 Extension Theorem
11.3 Compact and Complex Lie Algebras
11.3.1 Weyl Unitary Trick
11.3.2 Dynkin Diagrams
11.3.3 Cartan Subalgebras and Regular Elements
11.4 Maximal Tori
11.5 Center and Roots
11.6 Riemannian Geometry
11.7 Exercises
12 Noncompact Semi-Simple Groups
12.1 Cartan Decompositions
12.1.1 Cartan Decomposition of a Lie Algebra
12.1.2 Global Cartan Decomposition
12.2 Iwasawa Decompositions
12.2.1 Iwasawa Decomposition of a Lie Algebra
12.2.2 Global Iwasawa Decomposition
12.3 Classification
12.4 Exercises
Part IV Transformation Groups
Overview
13 Lie Group Actions
13.1 Group Actions
13.1.1 Orbits
13.2 Lie–Palais Theorem
13.2.1 Families of Vector Fields
13.3 Bundles
13.3.1 Principal Bundles
13.3.2 Associated Bundles
13.4 Homogeneous Spaces and Bundles
13.5 Exercises
14 Invariant Geometry
14.1 Complex Manifolds
14.1.1 Complex Lie Groups
14.2 Differential Forms and de Rham Cohomology
14.3 Riemannian Manifolds
14.4 Symplectic Manifolds
14.4.1 Coadjoint Representation
14.4.2 Moment Map
14.5 Exercises
Part V Appendices
A Vector Fields and Lie Brackets
A.1 Exercises
B Integrability of Distributions
B.1 Immersions and Submanifolds
B.2 Characteristic Distributions
B.3 Maximal Integral Manifolds
B.4 Adapted Charts
B.5 Integral Manifolds Are Quasi-Regular
B.6 Exercises
References
Index
