**Graduate Studies in Mathematics**

Volume: 11;
1996;
379 pp;
Hardcover

MSC: Primary 17;

**Print ISBN: 978-0-8218-0560-2
Product Code: GSM/11**

List Price: $80.00

AMS Member Price: $64.00

MAA Member Price: $72.00

**Electronic ISBN: 978-1-4704-2229-5
Product Code: GSM/11.E**

List Price: $75.00

AMS Member Price: $60.00

MAA Member Price: $67.50

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# Enveloping Algebras

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*Jacques Dixmier*

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

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.

#### Reviews & Endorsements

For the graduate student, this is a masterpiece of pedagogical writing, being succinct, wonderfully self-contained and of exceptional precision.

-- Mathematical Reviews

Self-contained, written with precision and elegance … an excellent textbook for the graduate student, a very good background for the professional algebraist not very familiar with the subject and a very useful source of references for the expert.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Enveloping Algebras

- Cover Cover11
- Title v6
- Copyright vi7
- Contents vii8
- Preface xi12
- Preface to the English edition xvii18
- Notation xix20
- CHAPTER 1. Lie Algebras 122
- 1.1. General remarks 122
- 1.2. Representations 425
- 1.3. Solvable and nilpotent Lie algebras 1132
- 1.4. The radical. The largest nilpotent ideal 1738
- 1.5. Semi-simple Lie algebras 1940
- 1.6. Semi-simplicity of representations 2142
- 1.7. Reductive Lie algebras 2647
- 1.8. Representations of sl(2,k) 3152
- 1.9. Cartan subalgebras 3354
- 1.10. The system of roots of a split semi-simple Lie algebra 3758
- 1.11. Regular linear forms 4667
- 1.12. Polarizations 5071
- 1.13. Symmetric semi-simple Lie algebras 5778
- 1.14. Supplementary remarks 6283

- CHAPTER 2. Enveloping Algebras 6687
- 2.1. The Poincaré-Birkhoff-Witt theorem 6687
- 2.2. The functor U 7091
- 2.3. The filtration of the enveloping algebra 7596
- 2.4. The canonical mapping of the symmetric algebra into the enveloping algebra 7798
- 2.5. The existence of finite-dimensional representations 82103
- 2.6. The commutant of a simple module 85106
- 2.7. The dual of the enveloping algebra 89110
- 2.8. Supplementary remarks 96117

- CHAPTER 3. Two Sided Ideals in Enveloping Algebras 101122
- 3.1. Primitive ideals and prime ideals 101122
- 3.2. The space of primitive ideals 105126
- 3.3. The passage to an ideal of g 107128
- 3.4. Extension of the scalar field 111132
- 3.5. The Krull dimension 112133
- 3.6. Rings of fractions 117138
- 3.7. Prime ideals in the solvable case 125146
- 3.8. Supplementary remarks 128149

- CHAPTER 4. Centres 131152
- 4.1. Notation 131152
- 4.2. Centre and core in the semi-simple case 133154
- 4.3. The semi-centre 134155
- 4.4. Centre and core in the solvable case 135156
- 4.5. The characterization of primitive ideals in the solvable case 141162
- 4.6. Heisenberg and Weyl algebras 146167
- 4.7. Centre and core in the nilpotent case 151172
- 4.8. Invariant ideals of the symmetric algebra (the nilpotent case) 158179
- 4.9. Supplementary remarks 162183

- CHAPTER 5. Induced Representations 169190
- CHAPTER 6. Primitive Ideals (The Solvable Case) 192213
- 6.1. The ideals I(f) 192213
- 6.2. Rational ideals in the nilpotent case 199220
- 6.3. Prime ideals of the enveloping algebra and invariant prime ideals of the symmetric algebra (the nilpotent case) 204225
- 6.4. The Jacobson topology 207228
- 6.5. The injectivity of the mapping I 215236
- 6.6. Supplementary remarks 227248

- CHAPTER 7. Verma Modules 231252
- 7.0. Notation 231252
- 7.1. The modules L(λ) and M(λ) 232253
- 7.2. Finite-dimensional representations 236257
- 7.3. Invariants in the symmetric algebra 239260
- 7.4. The Harish-Chandra homomorphism 242263
- 7.5. Characters 246267
- 7.6. Submodules of M(λ) 249270
- 7.7. Submodules of M(λ) and the ordering relation on the Weyl group 264285
- 7.8. Supplementary remarks 267288

- CHAPTER 8. The Enveloping Algebra of a Semi-simple Lie Algebra 277298
- CHAPTER 9. Harish-Chandra Modules 295316
- 9.0. Notation 295316
- 9.1. The case of a Lie subalgebra which is reductive in g 295316
- 9.2. Canonical mappings defined by a symmetrizing subalgebra 301322
- 9.3. The principal series 308329
- 9.4. The subquotient theorem 310331
- 9.5. Finiteness theorems 313334
- 9.6. Spherical modules in the diagonal case 315336
- 9.7. Supplementary remarks 321342

- CHAPTER 10. Primitive Ideals (The General Case) 325346
- CHAPTER 11. Appendix 346367
- Problems 354375
- Bibliography 359380
- Supplementary Bibliography 366387
- Added in 1996 371392
- Subject Index 375396
- Back Cover Back Cover1401