This self-contained text is an excellent introduction to Lie groups and their actions on manifolds. The authors start with an elementary discussion of matrix groups, followed by chapters devoted to the basic structure and representation theory of finite dimensinal Lie algebras. They then turn to global issues, demonstrating the key issue of the interplay between differential geometry and Lie theory. Special emphasis is placed on homogeneous spaces and invariant geometric structures. The last section of the book is dedicated to the structure theory of Lie groups. Particularly, they focus on maximal compact subgroups, dense subgroups, complex structures, and linearity.
This text is accessible to a broad range of mathematicians and graduate students; it will be useful both as a graduate textbook and as a research reference.
สารบัญ
Preface.- 1 Introduction.- Part I Matrix Groups.- 2 Concrete Matrix Groups.- 3 The Matrix Exponential Function.- 4 Linear Lie Groups.- Part II Lie Algebras.- 5 Elementary Structure Theory of Lie Algebras.- 6 Root Decomposition.- 7 Representation Theory of Lie Algebras.- Part III Manifolds and Lie Groups.- 8 Smooth Manifolds.- 9 Basic Lie Theory.- 10 Smooth Actions of Lie Groups.- Part IV Structure Theory of Lie Groups.- 11 Normal Subgroups, Nilpotemt and Solvable Lie Groups.- 12 Compact Lie Groups.- 13 Semisimple Lie Groups.- 14 General Structure Theory.- 15 Complex Lie Groups.- 16 Linearity of Lie Groups.- 17 Classical Lie Groups.- 18 Nonconnected Lie Groups.- Part V Appendices.- A Basic Covering Theory.- B Some Multilinear Algebra.- C Some Functional Analysis.- D Hints to Exercises.- References.- Index.
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