Graduate Algebra: Noncommutative View

Introduction xiii
List of symbols xvii
Prerequisites xxiii


Part IV. The Structure of Rings 1

Introduction 3

Chapter 13. Fundamental Concepts in Ring Theory 5
Matrix rings 7
Basic notions for noncommutative rings 14
Direct products of rings 16
The structure of Hom(M,N) 19
Representations of rings and algebras 21
The regular representation of an algebra 25
Supplement: Polynomial rings 26
Appendix 13A. Ring constructions using the regular representation 28

Chapter 14. Semisimple Modules and Rings and the
Wedderburn-Artin Theorem 33
Semisimple modules 33
Semisimple rings 37
The Wedderburn-Artin Theorem 40
Supplement: Rings with involution 43

Chapter 15. The Jacobson Program Applied to Left Artinian Rings 45
Primitive rings and ideals 46
The Jacobson radical 50
The structure of left Artinian rings 50
Supplement: The classical theory of finite-dimensional algebras 54
Appendix 15A: Structure theorems for rings and algebras 55
Appendix 15B. Kolchin’s Theorem and the Kolchin Problem 60

Chapter 16. Noetherian Rings and the Role of Prime Rings 63
Prime rings 64
Rings of fractions and Goldie’s Theorems 67
Applications to left Noetherian rings 77
The representation theory of rings and algebras: An introduction 78
Supplement: Graded and filtered algebras 82
Appendix 16A: Deformations and quantum algebras 83

Chapter 17. Algebras in Terms of Generators and Relations 87
Free algebraic structures 88
The free group 93
Resolutions of modules 99
Graphs 100
Growth of algebraic structures 104
Gel
fand-Kirillov dimension 109
Growth of groups 114
Appendix 17A. Presentations of groups 121
Groups as fundamental groups 122
Appendix 17B. Decision problems and reduction procedures 124
Appendix 17C: An introduction to the Burnside Problem 134

Chapter 18. Tensor Products 137
The basic construction 138
Tensor products of algebras 147Contents ix
Applications of tensor products 150
Exercises – Part IV 161
Chapter 13 161
Appendix 13A 164
Chapter 14 165
Chapter 15 167
Appendix 15A 170
Appendix 15B 171
Chapter 16 173
Appendix 16A 179
Chapter 17 180
Appendix 17A 184
Appendix 17B 187
Appendix 17C 187
Chapter 18 189


Part V. Representations of Groups and Lie Algebras 193

Introduction 195

Chapter 19. Group Representations and Group Algebras 197
Group representations 197
Modules and vector spaces over groups 202
Group algebras 204
Group algebras over splitting fields 211
The case when F is not a splitting field 216
Supplement: Group algebras of symmetric groups 218
Appendix 19A. Representations of infinite groups 228
Linear groups 230
Appendix 19B: Algebraic groups 238
The Tits alternative 244

Chapter 20. Characters of Finite Groups 249
Schur’s orthogonality relations 250
The character table 254
Arithmetic properties of characters 257
Tensor products of representations 260
Induced representations and their characters 263

Chapter 21. Lie Algebras and Other Nonassociative Algebras 271
Lie algebras 273
Lie representations 278
Nilpotent and solvable Lie algebras 282
Semisimple Lie algebras 288
The structure of f.d. semisimple Lie algebras 293
Cartan subalgebras 296
Lie structure in terms of sl(2, F) 301
Abstract root systems 307
Cartan’s classification of semisimple Lie algebras 311
Affine Lie algebras 316
Appendix 21A. The Lie algebra of an algebraic group 320
Appendix 21B: General introduction to nonassociative algebras 321
Some important classes of nonassociative algebras 323
Appendix 21C: Enveloping algebras of Lie algebras 331

Chapter 22. Dynkin Diagrams (Coxeter-Dynkin Graphs and
Coxeter Groups) 337
Dynkin diagrams 338
Reflection groups 346
A categorical interpretation of abstract Coxeter graphs 349
Exercises – Part V 355
Chapter 19 355
Appendix 19A 360
Appendix 19B 365
Chapter 20 368
Chapter 21 371
Appendix 21A 383
Appendix 21B 385
Appendix 21C 391
Chapter 22 394


Part VI. Representable Algebras 401

Introduction 403

Chapter 23. Polynomial Identities and Representable Algebras 405
Identities of finite-dimensional algebras 409
Central polynomials 413
The Grassmann algebra 416
Main theorems in PI-structure theory 417
Varieties and relatively free algebras 423
PI-theory and the symmetric group 428
Appendix 23A: Affine PI-algebras 429
Kemer’s solution of Specht’s conjecture in characteristic 0 434
Appendix 23B: Identities of nonassociative algebras 439
Identities of Lie algebras and the Restricted Burnside Problem 440

Chapter 24. Central Simple Algebras and the Brauer Group 447
Basic examples 448
The Brauer group 451
Subfields and centralizers 455
Division algebras described in terms of maximal subfields 460
The exponent 468
Techniques generalized from field theory 471
Galois descent and the corestriction map 474
Central simple algebras over local fields 478
Appendix 24A: Csa’s and geometry 482
Appendix 24B: Infinite-dimensional division algebras 484

Chapter 25. Homological Algebra and Categories of Modules 485
Exact and half-exact functors 487
Projective modules 491
Injective modules 500
Homology and cohomology 501
δ-functors and derived functors 509
Examples of homology and cohomology 516
Appendix 25A: Morita’s theory of categorical equivalence 523
Appendix 25B: Separable algebras 530
Azumaya algebras 534
Appendix 25C: Finite-dimensional algebras revisited 538

Chapter 26. Hopf Algebras 547
Coalgebras and bialgebras 547
Hopf modules 553
Quasi-triangular Hopf algebras and the quantum
Yang-Baxter equations (QYBEs) 556
Finite-dimensional Hopf algebras 559
Exercises – Part VI 563
Chapter 23 563
Appendix 23A 569
Appendix 23B 569
Chapter 24 572
Appendix 24A 579
Chapter 25 581
Appendix 25A 589
Appendix 25B 591
Appendix 25C 593
Chapter 26 594

List of major results 599

Bibliography 627

List of names 635

Index 637