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Lie groups. (English) Zbl 0955.22001

Universitext. Berlin: Springer. viii, 344 p. (2000).
This book is a (post)graduate textbook on Lie groups and Lie algebras (the finite-dimensional case). It is devoted to an exposition of this theory with an emphasis on differential-geometrical methods, in the spirit of S. Lie himself. The book consists of four chapters. Chapter 1 treats the fundamental properties of Lie groups and Lie algebras and their relations. It ends with a demonstration of Lie’s third fundamental theorem (on the existence of a simply connected Lie group with a prescribed finite-dimensional Lie algebra). Chapter 2 is devoted to proper actions of groups on manifolds (the stratification of the manifold into orbit types etc., in this case the orbit space is Hausdorff). In Chapter 3 the structure of compact Lie groups is studied in terms of the action of the group on itself by conjugation. Chapter 4 treats representations of compact groups. It has two focal points: the first one is the Peter-Weyl theorem for general compact groups, the second one is the classification of the representations of compact Lie groups: Weyl’s formula for characters, Cartan’s highest weight theorem, the Borel-Weil theorem (on the realization of representations on sections of holomorphic line bundles over flag manifolds). The nonconnected case is also considered. Two appendices contain a review of the required background material (some notions from differential geometry and some basic facts on ordinary differential equations on manifolds). One more quality of the book is the exercises concluding each chapter. The book is very useful both for beginners and for experts.

MSC:

22-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups
22E15 General properties and structure of real Lie groups
22E10 General properties and structure of complex Lie groups
22E30 Analysis on real and complex Lie groups
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22E46 Semisimple Lie groups and their representations
22E60 Lie algebras of Lie groups
22E67 Loop groups and related constructions, group-theoretic treatment
43A80 Analysis on other specific Lie groups
53C30 Differential geometry of homogeneous manifolds
57S25 Groups acting on specific manifolds
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