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Elements of mathematics. Lie groups and Lie algebras. Chapters 4–6. Reprint of the 1968 original. (Éléments de mathématique. Groupes et algèbres de Lie. Chapitres 4 à 6.) (French) Zbl 1120.17002
Berlin: Springer (ISBN 978-3-540-34490-2/pbk). 288 p. (2007).
The volume under review is an unabridged reprint of the Chapters 4, 5 and 6 of Book 9 of Nicolas Bourbaki’s fundamental treatise “Éléments de Mathématique”, following the French original edition published in 1968 by S. A. Masson, Paris, France. Being one of the later parts of Bourbaki’s work, Book 9 provides a comprehensive exposition of the modern abstract theory of Lie algebras and Lie groups, thereby methodologically building upon the algebraic, analytic, and topological structures developed in the first six, predominantiy foundational books of this sweeping treatise. After a systematic and thorough explanation of the basic theory of Lie algebras and Lie groups in the foregoing Chapters 1–3, including their fundamental notions and structural properties, their general representation theory, and their crucial interrelations in great detail, the subsequent four chapters under review turn to the more advanced and more refined topics of the subject. As the 1968 French original (Zbl 0186.33001) as well as its Russian translation (Zbl 0249.22001) and its English translation (Zbl 0983.17001) have already been analyzed by three different reviewers in the past, we certainly may confine ourselves to recall the main contents of these important, more specific Chapters 4–6 of Bourbaki’s book on Lie algebras and Lie groups.
Chapter 4 is devoted to the topic of Coxeter groups and Tits systems, together with the related background material from the theory of graphs and trees, whereas Chapter 5 comprehensively develops the structure theory of linear algebraic (Lie) groups generated by reflections. The latter topic naturally includes the simplicial and combinatorial aspects of these particular groups as well as the geometric represention theory of Coxeter groups and their invariant theory.
Chapter 6, one of the most important and most quoted parts of Bourbakits work as a whole, treats the framework of root systems within the classification theory of Lie algebras in a very elegant, systematic and widely exhaustive exposition. General root systems, reduced root systems, greatest roots, fundamental weights, dominant weights, affine Weyl groups, exponential invariants, and the complete classification of root systems via their Dynkin diagrams are among the main concepts and results constituting this chapter.
Each of these three chapters is enriched by a huge amount of exercises of the notorious, already almost legendary Bourbaki style, thereby providing a wealth of additional concepts, methods, and results with respect of the topics discussed in the present volume. The classification of root systems is compiled and illustrated by ten synoptical tables at the end of the volume, followed by an extensive summary of the principal properties of root systems as developed in the book. Also, there is a set of historical notes to Chapters 4, 5 and 6, surprisingly (for Bourbaki standards) accompanied by a list of about thirty bibliographical references.
Apparently, Jacques Tits had a great influence on the text in its present form, as can be guessed from the fact that Bourbaki distinctly acknowledges his precious help in the preface to the present volume.

##### MSC:
 17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras 22-02 Research exposition (monographs, survey articles) pertaining to topological groups 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 20E42 Groups with a $$BN$$-pair; buildings 20F55 Reflection and Coxeter groups (group-theoretic aspects) 51F15 Reflection groups, reflection geometries 01A72 Schools of mathematics
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