Handbook of Algebra (eBook)

Front Cover 1
Handbook of Algebra 4
Copyright Page 5
Contents 24
Preface 6
Outline of the Series 10
List of Contributors 24
Section 1A: Linear Algebra 28
Chapter 1. Matrix Invariants over Semirings 30
1. Introduction 31
2. Matrices and determinants 32
3. Semimodules: bases and dimension 37
4. Rank functions 41
5. Relations between different rank functions 45
6. Arithmetic behavior of rank 49
Acknowledgments 58
References 58
Chapter 2. Quadratic Forms 62
1. Introduction 63
2. Quadratic forms over fields 64
3. Witt rings of fields 66
4. Hasse and Witt invariants 70
5. Milnor’s Conjecture 72
6. Classifcation 74
7. Function fields 80
8. Sums of squares 85
9. Witt rings of rings 88
10. AbstractWitt rings 92
11. Historical 96
References 98
Section 2B: Homological Algebra. Cohomology. Cohomological Methods in Algebra. Homotopical Algebra 108
Chapter 3. Crossed Complexes and Higher Homotopy Groupoids as Noncommutative Tools for Higher Dimensional Local-to-Global Problems 110
Introduction 112
1. Crossed modules 114
2. The fundamental groupoid on a set of base points 116
3. The search for higher homotopy groupoids 119
4. Main results 125
5. Why crossed complexes? 127
6. Why cubical ?-groupoids with connections? 128
7. The equivalence of categories 129
8. First main aim of the work: higher Homotopy van Kampen theorems 130
9. The fundamental cubical ?-groupoid ?X* of a filtered space X* 131
10. Collapsing 133
11. Partial boxes 134
12. Thin elements 135
13. Sketch proof of the HHvKT 135
14. Tensor products and homotopies 137
15. Free crossed complexes and free crossed resolutions 139
16. Classifying spaces and the homotopy classification of maps 140
17. Relation with chain complexes with a groupoid of operators 141
18. Crossed complexes and simplicial groups and groupoids 143
19. Other homotopy multiple groupoids 144
20. Conclusion and questions 145
References 146
Section 4H: Hopf Algebras and Related Structures 152
Chapter 4. Hopf Algebraic Approach to Picard–Vessiot Theory 154
Introduction 155
Part I: PV theory in the differential context 157
Part II: PV theory in the C-ferential context 176
Part III: Unified PV theory 186
References 197
Chapter 5. Hopf Algebroids 200
1. Introduction 201
2. R-rings and R-corings 203
3. Bialgebroids 209
4. Hopf algebroids 231
Acknowledgment 259
References 259
Chapter 6. Comodules and Corings 264
1. Introduction 265
2. Categorical preliminaries 266
3. Corings 271
4. Comodules 281
5. Special types of corings and comodules 295
6. Applications 315
7. Extensions and dualizations 328
Acknowledgments 338
References 338
Chapter 7. Witt vectors. Part 1 346
1. Introduction and delimitation 352
2. Terminology 356
3. The p-adic Witt vectors. More historical motivation 356
4. Teichmüller representatives 358
5. Construction of the functor of the p-adicWitt vectors 358
6. The ring of p-adic Witt vectors over a perfect ring of characteristic p 365
7. Cyclic Galois extension of degree pn over a field of characteristic p 371
8. Cyclic central simple algebras of degree pn over a field of characteristic p 373
9. The functor of the big Witt vectors 375
10. The Hopf algebra Symm as the representing algebra for the bigWitt vectors 390
11. QSymm, the Hopf algebras of quasisymmetric functions and NSymm, the Hopf algebra of noncommutative symmetric functions 396
12. Free, cofree and duality properties of Symm 404
13. Frobenius and Verschiebung and other endomorphisms of A and the Witt vectors. 409
14. Supernatural and other quotients of the big Witt vectors 420
15. Cartier algebra and Dieudonné algebra 425
16. More operations on the and W functors: l-rings. 436
17. Necklace rings 454
18. Symm vs n R(Sn) 461
19. Burnside rings 469
Appendix. The algebra of symmetric functions in in.nitely many indeterminates 479
References 481
Chapter 8. Crystal Graphs and the Combinatorics of Young Tableaux 500
1. Introduction 501
2. Quantum group and crystal base 501
3. Crystals and Young tableaux 507
4. Crystal equivalence 513
5. Bicrystals 518
References 530
Section 6D: Representation Theory of Algebras 532
Chapter 9. Quivers and Representations 534
1. Introduction 536
2. Kac–Moody algebras and quantum groups 541
3. Quivers and their representations 546
4. Dimension vectors and positive roots 560
5. Ringel–Hall algebras 563
6. PBW basis and canonical basis 566
7. Root categories, Kac–Moody algebras and elliptic Lie algebras 577
8. Guide to the literature 586
Acknowledgements 586
References 586
Section 8: Applied Algebra 590
Chapter 10. Canonical Decompositions and Invariants for Data Analysis 592
1. Introduction 593
2. A general principle of interpretability 593
3. Symmetry studies 594
4. The regular invariants of S3 598
5. The regular invariants of S4 603
6. Comments and summary 609
Appendix A 610
References 611
Index 612