Supersymmetries and Infinite-Dimensional Algebras (eBook)

N. H. March (Herausgeber)

eBook Download: PDF

2013 | 1. Auflage
628 Seiten
Elsevier Science (Verlag)
978-1-4832-8837-6 (ISBN)

Infos

Recent devopments, particularly in high-energy physics, have projected group theory and symmetry consideration into a central position in theoretical physics. These developments have taken physicists increasingly deeper into the fascinating world of pure mathematics. This work presents important mathematical developments of the last fifteen years in a form that is easy to comprehend and appreciate.

Inhalt

Front Cover 1
Supersymmetries and Infinite-Dimensional Algebras 4
Copyright Page 5
Table of Contents 10
Preface 6
Contents of Volume I 16
Contents of Volume II 20
Part D: Lie Superalgebras, Lie Supergroups and their Applications 24
Chapter 20. Introduction to Superalgebras and Supermatrices 26
1 The notion of grading 26
2 Associative superalgebras 28
3 Grassmann algebras 30
4 Supermatrices 36
Chapter 21. General Properties of Lie Superalgebras 44
1 Lie superalgebras introduced 44
2 Definitions and immediate consequences 45
3 Subalgebras, direct sums and homomorphisms of Lie superalgebras 54
4 Graded representations of Lie superalgebras 58
5 The adjoint representation and the Killing form of a Lie superalgebra 64
Chapter 22. Superspace and Lie Supergroups 68
1 Grassmann variables as coordinates 68
2 Analysis on superspace 69
3 Linear Lie supergroups 91
Chapter 23. The Poincaré Superalgebras and Supergroups 102
1 Introduction 102
2 The N = 1, D = 4 Poincaré superalgebra and supergroup 103
3 Extended Poincaré superalgebras and Poincaré supergroups for D = 4 130
4 The Poincaré superalgebras and supergroups for Minkowski space-times of general dimension D 141
5 Irreducible representations of the unextended D = 4 Poincaré superalgebra 149
6 Irreducible representations of the extended D = 4 Poincaré superalgebras 161
7 Irreducible representations of the Poincaré superalgebras for general space-time dimensions 172
Chapter 24. Poincaré Supersymmetric Fields 184
1 Supersymmetric field theory 184
2 Supersymmetric multiplets 185
3 Superfields 204
4 Supersymmetric gauge theories 215
5 Spontaneous symmetry breaking 237
Chapter 25. Simple Lie Superalgebras 242
1 An outline of the presentation 242
2 The definition of a simple Lie superalgebra and some immediate consequences 243
3 Classical simple Lie superalgebras 246
4 Graded representations of basic classical simple complex Lie superalgebras 268
5 The classical simple real Lie superalgebras 290
6 The conformal, de Sitter and anti-de Sitter superalgebras 298
Part E: Infinite-Dimensional Lie Algebras and Superalgebras and their Applications 302
Chapter 26. The Structure of Kac-Moody Algebras 304
1 Introduction to infinite-dimensional Lie algebras 304
2 Construction of Kac-Moody algebras 305
3 Properties of general Kac-Moody algebras 312
4 Types of complex Kac-Moody algebras 322
5 Affine Kac-Moody algebras 330
6 Kac-Moody superalgebras 358
Chapter 27. Representations of Kac-Moody Algebras 362
1 Highest weight representations of general Kac-Moody algebras 362
2 Highest weight representations of affine Kac-Moody algebras 365
3 Character formulae 374
4 The vertex construction of the basic representation of a simply laced untwisted affine Kac-Moody algebra 376
5 Representations of untwisted affine Kac-Moody algebras in terms of fermion creation and annihilation operators 385
Chapter 28. The Virasoro Algebra and Superalgebras 392
1 The conformal algebras 392
2 Representations of the Virasoro algebra 396
3 Some constructions of highest weight representations of the Virasoro algebra 399
4 Virasoro superalgebras 405
Chapter 29. Algebraic Aspects of the Theory of Strings and Superstrings 412
1 Introduction 412
2 The bosonic string 412
3 The spinning string of Ramond, Neveu and Schwarz 434
4 The superstring of Green and Schwarz 440
5 The heterotic string 460
6 Further developments 469
Appendices 472
Appendix K: Proofs of Certain Theorems on Supermatrices and Lie Superalgebras 474
1 Proofs of Theorems I and IV of Chapter 20, Section 4 474
2 Proof of Theorem I of Chapter 21, Section 4 480
3 Proof of Theorem III of Chapter 21, Section 5 482
4 Proofs of Theorems II, III, IV and V of Chapter 25, Section 2 483
5 Proofs of Theorems VI, VII, VIII, IX, XVI, XX, XXII and XXIII of Chapter 25, Sections 3(a), 3(b) and 3(c) 486
6 Proofs of Theorems III and IV of Chapter 25, Section 4(a) 493
Appendix L: Clifford Algebras 498
1 The Clifford algebras of D-dimensional space–times 498
2 Irreducible representations for the case in which D is even 500
3 Irreducible representations for the case in which D is odd 512
4 Connections between representations of the D-dimensional Minkowski Clifford algebra and those of the (D — 2)-dimensional Euclidean Clifford algebra 519
5 A matrix identity for the D = 4 Minkowski Clifford algebra 524
Appendix M: Properties of the Classical Simple Complex Lie Superalgebras 526
1 The basic type 1 classical simple complex Lie superalgebras A(r/s), r > s = 0
2 The basic type I classical simple complex Lie superalgebras A(r/r),r = 1 532
3 The basic type II classical simple complex Lie superalgebras B(r/s), r = 0,s = 1 536
4 The basic type I classical simple complex Lie superalgebras C(s), s = 2 544
5 The basic type II classical simple complex Lie superalgebras D(r/s), r = 2, s = 1 548
6 The basic type II classical simple complex Lie superalgebras D(2/1 a), with a a complex parameter taking all values other than 0, — 1 and 8
7 The basic type II classical simple complex Lie superalgebra F(4) 560
8 The basic type II classical simple complex Lie superalgebra G(3) 565
9 The strange type I classical simple complex Lie superalgebras P(r), r = 2 568
10 The strange type II classical simple complex Lie superalgebras Q(r), r = 2 570
Appendix N: Properties of the Complex Affine Kac-Moody Algebras 574
1 The complex untwisted affine Kac-Moody algebra A1(1) 574
2 The complex untwisted affine Kac-Moody algebras Al(1), l = 2 575
3 The complex untwisted affine Kac-Moody algebras Bl(1), l = 3 576
4 The complex untwisted affine Kac-Moody algebras C(1), l = 2 577
5 The complex untwisted affine Kac-Moody algebras D(1)l' l = 4 579
6 The complex untwisted affine Kac-Moody algebra E(1)6 581
7 The complex untwisted affine Kac-Moody algebra E(1)7 582
8 The complex untwisted affine Kac-Moody 
583 
9 The complex untwisted affine Kac-Moody algebra F4(1) 584
10 The complex untwisted affine Kac-Moody algebra G(1)2 585
11 The complex twisted affine Kac-Moody algebra .(2)2 586
12 The complex twisted affine Kac-Moody algebras .(2)2l' l = 2 587
13 The complex twisted affine Kac-Moody algebras A(2)2l-1' l = 3 589
14 The complex twisted affine Kac-Moody algebras D(2)l+1' l = 2 591
15 The complex twisted affine Kac-Moody algebra E6(2) 593
16 The complex twisted affine Kac-Moody algebra D4(3) 594
Appendix O: Proofs of Certain Theorems on Kac-Moody and Virasoro Algebras 598
1 Proofs of Theorems I, III and IV of Chapter 27, Section 2 598
2 Proofs of Theorems I and II of Chapter 27, Section 4 602
3 Proof of Theorem I of Chapter 27, Section 5 612
4 Proofs of Theorems I and 11 of Chapter 28, Section 3 614
References 622
Subject Index 640

Produkt-Details

• Erscheint lt. Verlag: 22.10.2013
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Wirtschaft ► Betriebswirtschaft / Management ► Unternehmensführung / Management
• ISBN-10: 1-4832-8837-4 / 1483288374
• ISBN-13: 978-1-4832-8837-6 / 9781483288376