Lie Algebras and Algebraic Groups

1. Front Matter

   Pages i-xvi

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2. Results on topological spaces

   Pages 1-9

3. Rings and modules

   Pages 11-29

4. Integral extensions

   Pages 31-37

5. Factorial rings

   Pages 39-53

6. Field extensions

   Pages 55-73

7. Finitely generated algebras

   Pages 75-86

8. Gradings and filtrations

   Pages 87-94

9. Inductive limits

   Pages 95-101

10. Sheaves of functions

    Pages 103-112

11. Jordan decomposition and some basic results on groups

    Pages 113-129

12. Algebraic sets

    Pages 131-145

13. Prevarieties and varieties

    Pages 147-166

14. Projective varieties

    Pages 167-182

15. Dimension

    Pages 183-190

16. Morphisms and dimension

    Pages 191-204

17. Tangent spaces

    Pages 205-218

18. Normal varieties

    Pages 219-232

19. Root systems

    Pages 233-275

20. Lie algebras

    Pages 277-297

The theory of groups and Lie algebras is interesting for many reasons. In the mathematical viewpoint, it employs at the same time algebra, analysis and geometry. On the other hand, it intervenes in other areas of science, in particularindi?erentbranchesofphysicsandchemistry.Itisanactivedomain of current research. Oneofthedi?cultiesthatgraduatestudentsormathematiciansinterested in the theory come across, is the fact that the theory has very much advanced, andconsequently,theyneedtoreadavastamountofbooksandarticlesbefore they could tackle interesting problems. One of the goals we wish to achieve with this book is to assemble in a single volume the basis of the algebraic aspects of the theory of groups and Lie algebras. More precisely, we have presented the foundation of the study of ?nite-dimensional Lie algebras over an algebraically closed ?eld of characteristic zero. Here, the geometrical aspect is fundamental, and consequently, we need to use the notion of algebraic groups. One of the main di?erences between this book and many other books on the subject is that we give complete proofs for the relationships between algebraic groups and Lie algebras, instead of admitting them. We have also given the proofs of certain results on commutative al- bra and algebraic geometry that we needed so as to make this book as se- contained as possible. We believe that in this way, the book can be useful for both graduate students and mathematicians working in this area. Let us give a brief description of the material treated in this book.

• Abstract algebra
• Algebra
• Algebraic geometry
• Algebraic groups
• Commutative algebra algebra algebras
• Zariski topology
• algebraic varieties

From the reviews:

"As Tauvel and Yu focus on algebraic groups, they approach Lie theory via algebraic geometry and even develop that subject from scratch … . For the purpose at hand, Tauvel and Yu’s work compares favorably … . Summing Up: Highly recommended. Upper-division undergraduates through professionals." (D.V. Feldman, Choice, 43:10, June 2006)

"The sheer volume of material covered herein should make this book an invaluable reference for people interested in, or teaching, Lie algebras or algebraic groups. It truly provides ‘one stop shopping’ for someone needing a result or hard-to-find proof. … I cannot even begin to imagine how much work must have gone into creating such a thorough and comprehensive reference, and I have no doubt it will be an important and useful addition to the literature on this subject." (Mark Hunacek, The Mathematical Gazette, 90:19, 2006)

"The focus of this book is the study of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero. … the book is largely self-contained. … the authors are extremely knowledgeable in their subjects and the reader can profit from the wealth of material contained in this book. Therefore this book is an ideal reference source and research guide for graduate students and mathematicians working in this area." (Benjamin Cahen, Zentralblatt MATH, Vol. 1068, 2005)

"The theory of Lie algebras and algebraic groups has been an area of active research for the last 50 years. … The aim of this book is to assemble in a single volume the algebraic aspects of the theory, so as to present the foundations of the theory in characteristic zero. Detailed proofs are included and some recent results are discussed in the final chapters. All the prerequisites on commutative algebra and algebraic geometry are included." (L’Enseignement Mathematique, Vol. 51 (3-4), 2006)

"This introduction to Lie algebras andalgebraic groups aims to provide a full background to the subject. … The book has an encyclopedic character, offering much else besides the actual subject." (Mathematika, Vol. 52, 2005)

"The stated goal of the authors is to provide a ‘foundation for the study of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero’ in a self-contained work that will be useful to ‘both graduate students and mathematicians working in this area’. … the book contains a wealth of detail and takes the reader from the basic classical concepts to the modern borders of this still-active area. Complete proofs are given and the authors present their material clearly and concisely throughout." (Duncan Melville, MathDL, March, 2006)

"This book offers ... complete presentation of the theory of the topics in its title over an algebraically closed field of characteristic zero. Assuming only an undergraduate background in abstract algebra, it covers in detail all the prerequisites that one needs for the theory of Lie algebras and algebraic groups together with the foundations of that theory. ... The book is well written and easy to follow ... ." (William M. McGovern, SIAM Reviews, Vol. 48 (1), 2006)

"The theory of algebraic groups and Lie algebras is a deeply advanced and developed area of modern mathematics. … The text is clearly written and the material is well organized and considered, so the present book may be strongly recommended both to a beginner looking for a self-contained introduction to the theory of algebraic groups and Lie algebras, and to a specialist who wants to have a systematic presentation of the theory." (Ivan V. Arzhantsev, Mathematical Reviews, Issue, 2006 c)

• Département de Mathématiques, Université de Poitiers, Boulevard Marie et Pierre Curie, Futuroscope Chasseneuil cedex, France

  Patrice Tauvel, Rupert W. T. Yu

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