ELEMENTS OF MATHEMATICS
NICOLAS BOURBAKI ELEMENTS OF MATHEMATICS
Lie Groups and Lie Algebras Chapters 7–9
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Originally published as GROUPES ET ALGÈBRES DE LIE © N.Bourbaki Translator Andrew Pressley Department of Mathematics King’s College London Strand, London WC2R 2LS United Kingdom e-mail:
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Mathematics Subject Classification (2000): 17B10, 17B15, 17-01, 22C05, 22E60, 22E46 Library of Congress Control Number: 2004111709
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CONTENTS
CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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CHAPTER VII CARTAN SUBALGEBRAS AND REGULAR ELEMENTS § 1. Primary decomposition of linear representations . . . . . . . . 1. Decomposition of a family of endomorphisms . . . . . . . . . . . . . 2. The case of a linear family of endomorphisms . . . . . . . . . . . . 3. Decomposition of representations of a nilpotent Lie algebra 4. Decomposition of a Lie algebra relative to an automorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Invariants of a semi-simple Lie algebra relative to a semi-simple action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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§ 2. Cartan subalgebras and regular elements of a Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Cartan subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Regular elements of a Lie algebra . . . . . . . . . . . . . . . . . . . . . . . 3. Cartan subalgebras and regular elements . . . . . . . . . . . . . . . . 4. Cartan subalgebras of semi-simple Lie algebras . . . . . . . . . . .
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§ 3. Conjugacy theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Elementary automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Conjugacy of Cartan subalgebras . . . . . . . . . . . . . . . . . . . . . . . 3. Applications of conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Conjugacy of Cartan subalgebras of solvable Lie algebras . . 5. Lie group case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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§ 4. Regular elements of a Lie group . . . . . . . . . . . . . . . . . . . . . . . . 1. Regular elements for a linear representation . . . . . . . . . . . . . . 2. Regular elements of a Lie group . . . . . . . . . . . . . . . . . . . . . . . . 3. Rel