Synopses & Reviews
This book is a (post)graduate textbook on Lie groups and Lie algebras. Its aim is to give a broad introduction to the field with an emphasis on using differential-geometrical methods, in the spirit of Lie himself. The structure of compact Lie groups is analyzed in terms of the action of the group on itself by conjugation. The book culminates in the classification of the representations of compact Lie groups and in their realization as sections of holomorphic line bundles over flag manifolds. The relations with algebraic and analytic models are also discussed. A review of the required background material is provided in appendices.
Synopsis:
This book on Lie groups and Lie algebras gives a broad introduction to the field with an emphasis on using differential-geometrical methods, in the spirit of Lie himself. Relations with algebraic and analytic models are discussed and a review of the required background material is provided in appendices. The authors are leading specialists in group theory and this volume has been long awaited.
Table of Contents
Preface.- Lie Groups and Lie Algebras: Lie Groups and their Lie Algebras; Examples; The Exponential Map; The Exponential Map for a Vector Space; The Tangent Map of Exp; The Product in Logarithmic Coordinates; Dynkin's Formula; Lie's Fundamental Theorems; The Component of the Identity; Lie Subgroups and Homomorphisms; Quotients; Connected Commutative Lie Groups; Simply Connected Lie Groups; Lie's Third Fundamental Theorem in Global Form; Exercises.- Proper Actions: Review; Bochner's Linearization Theorem; Slices; Associated Fiber Bundles; Smooth Functions on the Orbit Space; Orbit Types and Local Action Types; The Stratification by Orbit Types; Principal and Regular Orbits; Blowing Up.- Compact Lie Groups: Centralizers; The Adjoint Action; Connectedness of Centralizers; The Group of Rotations and its Covering Group;...... Representations of Compact Groups: ... Schur's Lemma; Averaging; The Peter-Weyl Theorem; .... Weight Exercises; The Borel-Weil Theorem.... .- Appendix.
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