Eléments de mathématique : groupes et algèbres de Lie, Chapitres 4 - 6, groupes de Coxeter et systèmes de Tits; groupes engendrés par des réflexions; systèmes de racines / N. Bourbaki

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. (Zentralblatt)