Introduction to the Theory of Lie Groups

1. Front Matter

   Pages i-ix

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2. Topological Groups

   • Roger Godement

   Pages 1-32

3. Simply Connected Spaces and Groups

   • Roger Godement

   Pages 33-60

4. Analytic Properties of Linear Groups

   • Roger Godement

   Pages 61-112

5. Manifolds and Lie Groups

   • Roger Godement

   Pages 113-140

6. The Lie Algebra of a Lie Group

   • Roger Godement

   Pages 141-227

7. The Exponential Map for Lie Groups

   • Roger Godement

   Pages 229-283

8. Back Matter

   Pages 285-293

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This textbook covers the general theory of Lie groups. By first considering the case of linear groups (following von Neumann's method) before proceeding to the general case, the reader is naturally introduced to Lie theory.


Written by a master of the subject and influential member of the Bourbaki group, the French edition of this textbook has been used by several generations of students. This translation preserves the distinctive style and lively exposition of the original. Requiring only basics of topology and algebra, this book offers an engaging introduction to Lie groups for graduate students and a valuable resource for researchers.

• Lie groups
• discrete groups
• linear groups
• locally compact groups
• MSC (2010): 22E15, 22E40, 58A40

“The book draws many very interesting connections, say to topological groups, that are rarely found in more recent books. Hence I believe it is a valuable source for material for courses on the topic.” (A. Cap, Monatshefte für Mathematik, Vol. 192 (4), August, 2020)

“This book is an educational text, in which practically all statements, which are contained in it, are proved. Therefore it is written for students, who want not only to study the theory of Lie groups and Lie algebras and to pass an examination, but to study the foundation of Lie theory for effectively using it in further scientific work.” (V. V. Gorbatsevich, zbMATH 1367.22001, 2017)

• Paris, France

  Roger Godement

Roger Godement (October 1, 1921 - July 21, 2016) is known for his work in functional analysis, and also his expository books. He started as a student at the École normale supérieure in 1940, where he became a student of Henri Cartan. He started research into harmonic analysis on locally compact abelian groups, finding a number of major results; this work was in parallel but independent of similar investigations in the USSR and Japan. Work on the abstract theory of spherical functions published in 1952 proved very influential in subsequent work, particularly that of Harish-Chandra. The isolation of the concept of square-integrable representation is attributed to him. The Godement compactness criterion in the theory of arithmetic groups was a conjecture of his. He later worked with Jacquet on the zeta function of a simple algebra. He was an active member of the Bourbaki group in the early 1950s, and subsequently gave a number of significant Bourbaki seminars. He also took part in the Cartan seminar. He also wrote texts on abstract algebra and mathematical analysis.

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