Lie Groups Beyond an Introduction

封面
Springer Science & Business Media, 2013年3月9日 - 608 頁
Fifty years ago Claude Chevalley revolutionized Lie theory by pub lishing his classic Theory of Lie Groups I. Before his book Lie theory was a mixture of local and global results. As Chevalley put it, "This limitation was probably necessary as long as general topology was not yet sufficiently well elaborated to provide a solid base for a theory in the large. These days are now passed:' Indeed, they are passed because Chevalley's book changed matters. Chevalley made global Lie groups into the primary objects of study. In his third and fourth chapters he introduced the global notion of ana lytic subgroup, so that Lie subalgebras corresponded exactly to analytic subgroups. This correspondence is now taken as absolutely standard, and any introduction to general Lie groups has to have it at its core. Nowadays "local Lie groups" are a thing of the past; they arise only at one point in the development, and only until Chevalley's results have been stated and have eliminated the need for the local theory. But where does the theory go from this point? Fifty years after Cheval ley's book, there are clear topics: E. Cartan's completion ofW. Killing's work on classifying complex semisimple Lie algebras, the treatment of finite-dimensional representations of complex semisimple Lie algebras and compact Lie groups by Cartan and H. Weyl, the structure theory begun by Cartan for real semisimple Lie algebras and Lie groups, and harmonic analysis in the setting of semisimple groups as begun by Cartan and Weyl.
 

內容

LIE ALGEBRAS AND LIE GROUPS 1 Definitions and Examples
1
Ideals
10
Field Extensions and the Killing Form
11
Semidirect Products of Lie Algebras
15
Solvable Lie Algebras and Lies Theorem
17
Nilpotent Lie Algebras and Engels Theorem
22
Cartans Criterion for Semisimplicity 8 Examples of Semisimple Lie Algebras
33
Representations of s2
37
31
243
37
248
43
249
55
254
58
255
62
257
66
261
73
263

Elementary Theory of Lie Groups
43
Automorphisms and Derivations
55
Semidirect Products of Lie Groups
58
Nilpotent Lie Groups
62
Classical Semisimple Lie Groups
66
Problems
73
COMPLEX SEMISIMPLE LIE ALGEBRAS
79
Classical Root Space Decompositions
80
Existence of Cartan Subalgebras
85
Uniqueness of Cartan Subalgebras
92
Roots
107
Abstract Root Systems
138
UNIVERSAL ENVELOPING ALGEBRA
164
COMPACT LIE GROUPS
181
Weyl Group
207
7
210
Classification of Abstract Cartan Matrices X
211
17
217
FINITEDIMENSIONAL REPRESENTATIONS
219
22
238
24
239
79
264
80
265
85
268
92
271
94
272
103
275
STRUCTURE THEORY OF SEMISIMPLE GROUPS
291
Isomorphism Theorem
341
Existence Theorem
344
ADVANCED STRUCTURE THEORY
372
INTEGRATION
456
APPENDICES
487
B Lies Third Theorem
504
Hints for Solutions of Problems
545
Problems
550
Notes
565
References
585
152
586
156
588
Index of Notation
595
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