Lie Groups

1. Front Matter

   Pages i-xiv

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2. Introduction

   • Luiz A. B. San Martin

   Pages 1-9

3. Topological Groups

   1. Front Matter

      Pages 11-12

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   2. Topological Groups

      • Luiz A. B. San Martin

      Pages 13-41

   3. Haar Measure

      • Luiz A. B. San Martin

      Pages 43-61

   4. Representations of Compact Groups

      • Luiz A. B. San Martin

      Pages 63-82

4. Lie Groups and Algebras

   1. Front Matter

      Pages 83-85

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   2. Lie Groups and Lie Algebras

      • Luiz A. B. San Martin

      Pages 87-116

   3. Lie Subgroups

      • Luiz A. B. San Martin

      Pages 117-143

   4. Homomorphisms and Coverings

      • Luiz A. B. San Martin

      Pages 145-162

   5. Series Expansions

      • Luiz A. B. San Martin

      Pages 163-176

5. Lie Algebras and Simply Connected Groups

   1. Front Matter

      Pages 177-179

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   2. The Affine Group and Semi-Direct Products

      • Luiz A. B. San Martin

      Pages 181-198

   3. Solvable and Nilpotent Groups

      • Luiz A. B. San Martin

      Pages 199-210

   4. Compact Groups

      • Luiz A. B. San Martin

      Pages 211-245

   5. Noncompact Semi-Simple Groups

      • Luiz A. B. San Martin

      Pages 247-263

6. Transformation Groups

   1. Front Matter

      Pages 265-266

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   2. Lie Group Actions

      • Luiz A. B. San Martin

      Pages 267-298

   3. Invariant Geometry

      • Luiz A. B. San Martin

      Pages 299-336

7. Back Matter

   Pages 337-371

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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.

• Lie groups
• Lie algebras
• Topological group
• Haar measure
• Homomorphism
• Enveloping algebras
• Compact groups
• Nilpotent
• Lie group action

“An important feature of the book is the presence of a lot of examples illustrating introduced concepts and proven results. Each chapter … accompanied by a fairly many exercises that enable the reader to check the degree of understanding of the material in each chapter and to learn something new. The student can use this book for self-study of the foundations of the theory of Lie groups.” (V. V. Gorbatsevich, zbMATH 1466.22001, 2021)

• Department of Mathematics—IMECC, State University of Campinas, Campinas, Brazil

  Luiz A. B. San Martin

Luiz Antonio Barrera San Martin is a Full Professor at the University of Campinas, Brazil. He holds a Master's degree in Mathematics (1982) from the University of Campinas, Brazil, and a PhD in Mathematics (1987) from the University of Warwick, England. His research interests are in Lie Theory, more precisely in semigroups, semisimple groups, Lie groups, homogeneous spaces, and flag manifolds.

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