This book, which is the first systematic exposition of the algebraic approach to representations of Lie groups via representations of (or modules over) the corresponding universal enveloping algebras, turned out to be so well written that even today it remains one of the main textbooks and reference books on the subject. In 1992, Jacques Dixmier was awarded the Leroy P. Steele Prize for expository writing in mathematics. The Committee's citation mentioned Enveloping Algebras as one of Dixmier's "extraordinary books". Written with unique precision and elegance, the book provides the reader with insight and understanding of this very important subject. It can be an excellent textbook for a graduate course, as well as a very useful source of references in the theory of universal enveloping algebras, the area of mathematics that remains as important today as it was 20 years ago.
For the 1996 printing, the author updated the status of open problems and added some relevant references.
Readership:Graduate students and research mathematicians interested in Lie algebras.
Author(s): Jacques Dixmier
Series: North-Holland mathematical library ; v. 14 (GSM/11)
Publisher: North-Holland (Elsevier) 1977 /AMS (1996)
Year: 1977 /1996
Language: English
Pages: C, XVI, 379, B
S Title
North-Holland Mathematical Library; Vol. 14
ENVELOPING ALGEBRAS
Copyright
(C) NORTH-HOLLAND PUBLISHING COMPANY - 1977
ISBN 0444110771
PREFACE
PREFACE TO THE ENGLISH EDITION
CONTENTS
NOTATION
CHAPTER 1 LIE ALGEBRAS
1.1. General remarks
1.2. Representations
1.3. Solvable and nilpotent Lie algebras
1.4. Tbe radical. The largest nilpotent ideal
1.5. Semi-simple Lie algebras
1.6. Semi-simplicity of representations
1.7. Reductive Lie algebras
1.8. Representations of g(l;k)
1.9. Cartan subalgebras
1.10. The system of roots of a split semi-simple Lie algebra
1.11. Regular linear forms
1.12. Polarizations
1.13. Symmetric semi-simple Lie algebras
1.14. Supplementary remarks
CHAPTER 2 ENVELOPING ALGEBRAS
2.1. The Poincare-Birkhoff-Witt theorem
2.2. The functor U
2.3. The filtration of the enveloping algebra
2.4. The canonical mapping of the symmetric algebra into the enveloping algebra
2.5. The existence of finite-dimensional representations
2.6. The commutant of a simple module
2. 7. The dual of the enveloping algebra
2.8. Supplementary remarks
CHAPTER 3 TWO SIDED IDEALS IN ENVELOPING ALGEBRAS
3.1. Primitive ideals and prime ideals
3.1. The space of primitive ideals
3.3. 'The passage to an ideal of g
3.4. Extension of the scalar field
3.5. Tbe Krull dimension
3.6. Rings of fracti01w
3.7. Prime ideals in the solvable case
3.8. Supplementary remarks
CHAPTER 4 CENTRES
4.1. Notation
4.2. Centre and core in the semi-simple case
4.3. The semi-centre
4.4. Centre and core in the solvable case
4.5. The characterization of primitive ideals in the solvable case
4.6. Heisenberg and Weyl algebras
4. 7. Centre and core in the nilpotent case
4.8. Invariant ideals of tbe symmetric algebra (the nilpotent case)
4.9. Supplementary remarks
CHAPTER 5 INDUCED REPRESENTATIONS
5.1. Induced representations
5.2. Twisted induced representations
5.3. A criterion for the simplicity of induced representations
5.4. The construction of primitive ideals by induction
5.5. Co-induced representadons
5.6. Supplementary remarks
CHAPTER 6 PRIMITIVE IDEALS (THE SOLVABLE CASE)
6.1. The ideals I(f)
6.2. Rational ideals in the nilpotent case
6.3. Prime ideals of the enveloping algebra and invariant prime ideals of the symmetric algebra (the nilpotent case)
6.4. The Jacobson topology
6.5. The injectivity of the mapping I
6.6. Supplementary remarks
CHAPTER 7 VERMA MODULES
7.0. Notation
7.1. The modules L(\lambda) and M(\lambda)
7.2. Finite-dimensional representations
7.3. Invariants in the symmetric algebra
7.4. The Jlarish..Chandra homomorphism
7.5. Characters
7.6. Submodules of M(\lambda.)
7.7. Submodules of M(A.) and the ordering relation on the Weyl group
7.8. Supplementary remarks
CHAPTER 8 THE ENVELOPING ALGEBRA OF A SEMI-SIMPLE LIE ALGEBRA
8.1. The cone of nilpotent elements
8.2. The enveloping algebra as a module over its centre
8.3. The adjoint representation in the enveloping algebra
8.4. The annihilators of Verma modules
8.5. Supplementary remarks
CHAPTER 9 HARISH-CHANDRA MODULES
9.0. Notation
9.1. The case of a Lie subalgebra which is reductive in g
9.2. Canonical mappings defined by a symmetrizing subalgebra
9.3. The principal series
9.4. The subquotient theorem
9.5. Finiteness theorems
9.6. Spherical modules in the diagonal case
9.7. Supplementary remarks
CHAPTER 10 PRIMITIVE IDEALS (THE GENERAL CASE)
10.1. Some canonical homomorphisms
10.2. Application to induced representations
10.3. The ideals I(f)
10.4. Application to the centre of the enveloping algebra
10.5. Supplementary remarks
CHAPTER 11 APPENDIX
11.1. Root systems
11.2. Miscellaneous results
PROBLEMS
PART I
PART II
PART III
BIBLIOGRAPHY
SUPPLEMENTARY BffiLIOGRAPHY
SUBJECT INDEX
Added in 1996
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