Lie Groups Beyond an IntroductionSpringer 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
1 | |
10 | |
11 | |
15 | |
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 |
585 | |
152 | 586 |
156 | 588 |
595 | |
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常見字詞
Ad(g Ad(k algebra g algebraically integral analytic subgroup automorphism basis Cartan decomposition Cartan involution Cartan matrix Cartan subalgebra Cartan subgroup Cayley transform commutes compact real form complex semisimple Lie complexification conjugate Corollary corresponding define diagonal dimension direct sum Dynkin diagram eigenvalues element example finite finite-dimensional representation follows formula Haar measure Hence Hermitian highest weight homomorphism imaginary induction inner product Int g invariant subspace isomorphism Killing form Lemma Let G Lie subalgebra linear map matrices maximal abelian subspace multiplication nilpotent nonzero one-one orthogonal parabolic subalgebra PROOF Proposition prove real Lie algebra reductive Lie group representation of g restricted roots root vectors scalar semisimple Lie algebra semisimple Lie group shows simple roots simply connected SO(n sp(n stable Cartan subalgebra subalgebra of go subgroup of G tensor Theorem V₁ vector space Vogan diagram Weyl group