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Lie Groups(Universitext) paper VIII, 344 p. 99

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発行年月	1999年12月
出版社／提供元	Springer-Verlag GmbH
出版国	ドイツ
言語	英語
媒体	冊子
装丁	paper
ページ数／巻数	VIII, 344 p.
ジャンル	洋書／理工学／数学／代数学
ISBN	9783540152934
商品コード	0209968636
本の性格	学術書
商品URL	https://kw.maruzen.co.jp/ims/itemDetail.html?itmCd=0209968636

内容

This book is devoted to an exposition of the theory of finite-dimensional Lie groups and Lie algebras, which is a beautiful and central topic in modern mathematics. At the end of the nineteenth century this theory came to life in the works of Sophus Lie. It had its origins in Lie's idea of applying Galois theory to differential equations and in Klein's "Erlanger Programm" of treat ing symmetry groups as the fundamental objects in geometry. Lie's approach to many problems of analysis and geometry was mainly local, that is, valid in local coordinate systems only. At the beginning of the twentieth century E. Cartan and Weyl began a systematic treatment of the global aspects of Lie's theory. Since then this theory has ramified tremendously and now, as the twentieth century is coming to a close, its concepts and methods pervade mathematics and theoretical physics. Despite the plethora of books devoted to Lie groups and Lie algebras we feel there is justification for a text that puts emphasis on Lie's principal idea, namely, geometry treated by a blend of algebra and analysis. Lie groups are geometrical objects whose structure can be described conveniently in terms of group actions and fiber bundles. Therefore our point of view is mainly differential geometrical. We have made no attempt to discuss systematically the theory of infinite-dimensional Lie groups and Lie algebras, which is cur rently an active area of research. We now give a short description of the contents of each chapter.

Preface 1 Lie Groups and Lie Algebras 1.1 Lie Groups and their Lie Algebras 1.2 Examples 1.3 The Exponential Map 1.4 The Exponential Map for a Vector Space 1.5 The Tangent Map of Exp 1.6 The Product in Logarithmic Coordinates 1.7 Dynkin's Formula 1.8 Lie's Fundamental Theorems 1.9 The Component of the Identity 1.10 Lie Subgroups and Homomorphisms 1.11 Quotients 1.12 Connected Commutative Lie Groups 1.13 Simply Connected Lie Groups 1.14 Lie's Third Fundamental Theorem in Global Form 1.15 Exercises 1.16 Notes References for Chapter One 2 Proper Actions 2.1 Review 2.2 Bochner's Linearization Theorem 2.3 Slices 2.4 Associated Fiber Bundles 2.5 Smooth Functions on the Orbit Space 2.6 Orbit Types and Local Action Types 2.7 The Stratification by Orbit Types 2.8 Principal and Regular Orbits 2.9 Blowing Up 2.10 Exercises 2.11 Notes References for Chapter Two 3 Compact Lie Groups 3.0 Introduction 3.1 Centralizers 3.2 The Adjoint Action 3.3 Connectedness of Centralizers 3.4 The Group of Rotations and its Covering Group 3.5 Roots and Root Spaces 3.6 Compact Lie Algebras 3.7 Maximal Tori 3.8 Orbit Structure in the Lie Algebra 3.9 The Fundamental Group 3.10 The Weyl Group as a Reflection Group 3.11 The Stiefel Diagram 3.12 Unitary Groups 3.13 Integration 3.14 The Weyl Integration Theorem 3.15 Nonconnected Groups 3.16 Exercises 3.17 Notes References for Chapter Three 4 Representations of Compact Groups 4.0 Introduction 4.1 Schur's Lemma 4.2 Averaging 4.3 Matrix Coefficients and Characters 4.4 G-types 4.5 Finite Groups 4.6 The Peter-Weyl Theorem 4.7 Induced Representations 4.8 Reality 4.9 Weyl's Character Formula 4.10 Weight Exercises 4.11 Highest Weight Vectors 4.12 The Borel-Weil Theorem 4.13 The Nonconnected Case 4.14 Exercises 4.15 Notes References for Chapter Four Appendix A Appendix B Appendix

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