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E-raamat: Lie Groups, Lie Algebras, and Representations: An Elementary Introduction

4.15/5 (29 hinnangut Goodreads-ist)

Brian Hall

Formaat: EPUB+DRM
Sari: Graduate Texts in Mathematics 222
Ilmumisaeg: 11-May-2015
Kirjastus: Springer International Publishing AG
Keel: eng
ISBN-13: 9783319134673

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Formaat: EPUB+DRM
Sari: Graduate Texts in Mathematics 222
Ilmumisaeg: 11-May-2015
Kirjastus: Springer International Publishing AG
Keel: eng
ISBN-13: 9783319134673

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This textbook treats Lie groups, Lie algebras and their representations in an elementary but fully rigorous fashion requiring minimal prerequisites. In particular, the theory of matrix Lie groups and their Lie algebras is developed using only linear algebra, and more motivation and intuition for proofs is provided than in most classic texts on the subject.

In addition to its accessible treatment of the basic theory of Lie groups and Lie algebras, the book is also noteworthy for including:

a treatment of the BakerCampbellHausdorff formula and its use in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebras motivation for the machinery of roots, weights and the Weyl group via a concrete and detailed exposition of the representation theory of sl(3;C) an unconventional definition of semisimplicity that allows for a rapid development of the structure theory of semisimple Lie algebras a self-contained construction of the representations of compact groups, independent of Lie-algebraic arguments

The second edition of Lie Groups, Lie Algebras, and Representations contains many substantial improvements and additions, among them: an entirely new part devoted to the structure and representation theory of compact Lie groups; a complete derivation of the main properties of root systems; the construction of finite-dimensional representations of semisimple Lie algebras has been elaborated; a treatment of universal enveloping algebras, including a proof of the PoincaréBirkhoffWitt theorem and the existence of Verma modules; complete proofs of the Weyl character formula, the Weyl dimension formula and the Kostant multiplicity formula.

Review of the first edition:

This is an excellent book. It deserves to, and undoubtedly will, become the standard text for early graduate courses in Lie group theory ... an important addition tothe textbook literature ... it is highly recommended.

The Mathematical Gazette

Arvustused

The first edition of this book was very good; the second is even better, and more versatile. This text remains one of the most attractive sources available from which to learn elementary Lie group theory, and is highly recommended. (Mark Hunacek, The Mathematical Gazette, Vol. 101 (551), July, 2017)

Part I General Theory

1 Matrix Lie Groups

(28)

1.1 Definitions

(2)

1.2 Examples

(11)

1.3 Topological Properties

(5)

1.4 Homomorphisms

(4)

1.5 Lie Groups

(1)

1.6 Exercises

(5)

2 The Matrix Exponential

(18)

2.1 The Exponential of a Matrix

(3)

2.2 Computing the Exponential

(2)

2.3 The Matrix Logarithm

(4)

2.4 Further Properties of the Exponential

(2)

2.5 The Polar Decomposition

(4)

2.6 Exercises

(3)

3 Lie Algebras

(28)

3.1 Definitions and First Examples

(4)

3.2 Simple, Solvable, and Nilpotent Lie Algebras

(2)

3.3 The Lie Algebra of a Matrix Lie Group

(2)

3.4 Examples

(3)

3.5 Lie Group and Lie Algebra Homomorphisms

(5)

3.6 The Complexification of a Real Lie Algebra

(2)

3.7 The Exponential Map

(3)

3.8 Consequences of Theorem 3.42

(3)

3.9 Exercises

(4)

4 Basic Representation Theory

(32)

4.1 Representations

(4)

4.2 Examples of Representations

(3)

4.3 New Representations from Old

(6)

4.4 Complete Reducibility

(4)

4.5 Schur's Lemma

(2)

4.6 Representations of sl(2; C)

(5)

4.7 Group Versus Lie Algebra Representations

101

(2)

4.8 A Nonmatrix Lie Group

103

(2)

4.9 Exercises

105

(4)

5 The Baker--Campbell--Hausdorff Formula and Its Consequences

109

(32)

5.1 The "Hard" Questions

109

(1)

5.2 An Illustrative Example

110

(3)

5.3 The Baker--Campbell--Hausdorff Formula

113

(1)

5.4 The Derivative of the Exponential Map

114

(3)

5.5 Proof of the BCH Formula

117

(1)

5.6 The Series Form of the BCH Formula

118

(1)

5.7 Group Versus Lie Algebra Homomorphisms

119

(7)

5.8 Universal Covers

126

(2)

5.9 Subgroups and Subalgebras

128

(7)

5.10 Lie's Third Theorem

135

(1)

5.11 Exercises

135

(6)

Part II Semisimple Lie Algebras

6 The Representations of sl(3; C)

141

(28)

6.1 Preliminaries

141

(1)

6.2 Weights and Roots

142

(4)

6.3 The Theorem of the Highest Weight

146

(2)

6.4 Proof of the Theorem

148

(5)

6.5 An Example: Highest Weight (1,1)

153

(1)

6.6 The Weyl Group

154

(4)

6.7 Weight Diagrams

158

(1)

6.8 Further Properties of the Representations

159

(6)

6.9 Exercises

165

(4)

7 Semisimple Lie Algebras

169

(28)

7.1 Semisimple and Reductive Lie Algebras

169

(5)

7.2 Cartan Subalgebras

174

(2)

7.3 Roots and Root Spaces

176

(6)

7.4 The Weyl Group

182

(1)

7.5 Root Systems

183

(2)

7.6 Simple Lie Algebras

185

(3)

7.7 The Root Systems of the Classical Lie Algebras

188

(5)

7.8 Exercises

193

(4)

8 Root Systems

197

(44)

8.1 Abstract Root Systems

197

(4)

8.2 Examples in Rank Two

201

(3)

8.3 Duality

204

(2)

8.4 Bases and Weyl Chambers

206

(6)

8.5 Weyl Chambers and the Weyl Group

212

(4)

8.6 Dynkin Diagrams

216

(2)

8.7 Integral and Dominant Integral Elements

218

(3)

8.8 The Partial Ordering

221

(7)

8.9 Examples in Rank Three

228

(4)

8.10 The Classical Root Systems

232

(4)

8.11 The Classification

236

(2)

8.12 Exercises

238

(3)

9 Representations of Semisimple Lie Algebras

241

(24)

9.1 Weights of Representations

242

(2)

9.2 Introduction to Verma Modules

244

(2)

9.3 Universal Enveloping Algebras

246

(4)

9.4 Proof of the PBW Theorem

250

(4)

9.5 Construction of Verma Modules

254

(3)

9.6 Irreducible Quotient Modules

257

(3)

9.7 Finite-Dimensional Quotient Modules

260

(3)

9.8 Exercises

263

(2)

10 Further Properties of the Representations

265

(42)

10.1 The Structure of the Weights

265

(4)

10.2 The Casimir Element

269

(4)

10.3 Complete Reducibility

273

(2)

10.4 The Weyl Character Formula

275

(6)

10.5 The Weyl Dimension Formula

281

(6)

10.6 The Kostant Multiplicity Formula

287

(7)

10.7 The Character Formula for Verma Modules

294

(1)

10.8 Proof of the Character Formula

295

(8)

10.9 Exercises

303

(4)

Part III Compact Lie Groups

11 Compact Lie Groups and Maximal Tori

307

(36)

11.1 Tori

308

(4)

11.2 Maximal Tori and the Weyl Group

312

(3)

11.3 Mapping Degrees

315

(6)

11.4 Quotient Manifolds

321

(5)

11.5 Proof of the Torus Theorem

326

(4)

11.6 The Weyl Integral Formula

330

(3)

11.7 Roots and the Structure of the Weyl Group

333

(6)

11.8 Exercises

339

(4)

12 The Compact Group Approach to Representation Theory

343

(28)

12.1 Representations

343

(3)

12.2 Analytically Integral Elements

346

(5)

12.3 Orthonormality and Completeness for Characters

351

(6)

12.4 The Analytic Proof of the Weyl Character Formula

357

(4)

12.5 Constructing the Representations

361

(5)

12.6 The Case in Which S is Not Analytically Integral

366

(3)

12.7 Exercises

369

(2)

13 Fundamental Groups of Compact Lie Groups

371

13.1 The Fundamental Group

371

(2)

13.2 Fundamental Groups of Compact Classical Groups

373

(4)

13.3 Fundamental Groups of Noncompact Classical Groups

377

(1)

13.4 The Fundamental Groups of K and T

377

(6)

13.5 Regular Elements

383

(6)

13.6 The Stiefel Diagram

389

(5)

13.7 Proofs of the Main Theorems

394

(5)

13.8 The Center of A

399

(4)

13.9 Exercises

403

Erratum

(442)

A Linear Algebra Review

407

(12)

A.1 Eigenvectors and Eigenvalues

407

(1)

A.2 Diagonalization

408

(1)

A.3 Generalized Eigenvectors and the SN Decomposition

409

(2)

A.4 The Jordan Canonical Form

411

(1)

A.5 The Trace

411

(1)

A.6 Inner Products

412

(2)

A.7 Dual Spaces

414

(1)

A.8 Simultaneous Diagonalization

415

(4)

B Differential Forms

419

(6)

C Clebsch--Gordan Theory and the Wigner--Eckart Theorem

425

(10)

C.1 Tensor Products of sl(2; C) Representations

425

(3)

C.2 The Wigner-Eckart Theorem

428

(4)

C.3 More on Vector Operators

432

(3)

D Completeness of Characters

435

(8)

References

443

(2)

Index

445

Brian Hall is Professor of Mathematics at the University of Notre Dame, IN.

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