Hazewinkel Handbook of Algebra

1. Auflage 2009
ISBN: 978-0-08-093281-1
Verlag: Elsevier Science & Technology
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, Band V6, 637 Seiten, Format (B × H): 165 mm x 240 mm

Reihe: Handbook of Algebra

Algebra, as we know it today, consists of many different ideas, concepts and results. A reasonable estimate of the number of these different items would be somewhere between 50,000 and 200,000. Many of these have been named and many more could (and perhaps should) have a name or a convenient designation. Even the nonspecialist is likely to encounter most of these, either somewhere in the literature, disguised as a definition or a theorem or to hear about them and feel the need for more information. If this happens, one should be able to find enough information in this Handbook to judge if it is worthwhile to pursue the quest.
In addition to the primary information given in the Handbook, there are references to relevant articles, books or lecture notes to help the reader. An excellent index has been included which is extensive and not limited to definitions, theorems etc.
The Handbook of Algebra will publish articles as they are received and thus the reader will find in this third volume articles from twelve different sections. The advantages of this scheme are two-fold: accepted articles will be published quickly and the outline of the Handbook can be allowed to evolve as the various volumes are published.
A particularly important function of the Handbook is to provide professional mathematicians working in an area other than their own with sufficient information on the topic in question if and when it is needed.
- Thorough and practical source of information
- Provides in-depth coverage of new topics in algebra
- Includes references to relevant articles, books and lecture notes

Autoren/Hrsg.

Hazewinkel, M.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Interdisziplinär Computeralgebra
• Mathematik | Informatik Mathematik Algebra Lineare und multilineare Algebra, Matrizentheorie
• Mathematik | Informatik Mathematik Algebra Homologische Algebra
• Mathematik | Informatik Mathematik Algebra Elementare Algebra
• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie

Inhaltsverzeichnis

1;Front Cover;1
2;Handbook of Algebra;4
3;Copyright Page;5
4;Contents;24
5;Preface;6
6;Outline of the Series;10
7;List of Contributors;24
8;Section 1A: Linear Algebra;28
8.1;Chapter 1. Matrix Invariants over Semirings;30
8.1.1;1. Introduction;31
8.1.2;2. Matrices and determinants;32
8.1.3;3. Semimodules: bases and dimension;37
8.1.4;4. Rank functions;41
8.1.5;5. Relations between different rank functions;45
8.1.6;6. Arithmetic behavior of rank;49
8.1.7;Acknowledgments;58
8.1.8;References;58
8.2;Chapter 2. Quadratic Forms;62
8.2.1;1. Introduction;63
8.2.2;2. Quadratic forms over fields;64
8.2.3;3. Witt rings of fields;66
8.2.4;4. Hasse and Witt invariants;70
8.2.5;5. Milnor’s Conjecture;72
8.2.6;6. Classifcation;74
8.2.7;7. Function fields;80
8.2.8;8. Sums of squares;85
8.2.9;9. Witt rings of rings;88
8.2.10;10. AbstractWitt rings;92
8.2.11;11. Historical;96
8.2.12;References;98
9;Section 2B: Homological Algebra. Cohomology. Cohomological Methods in Algebra. Homotopical Algebra;108
9.1;Chapter 3. Crossed Complexes and Higher Homotopy Groupoids as Noncommutative Tools for Higher Dimensional Local-to-Global Problems;110
9.1.1;Introduction;112
9.1.2;1. Crossed modules;114
9.1.3;2. The fundamental groupoid on a set of base points;116
9.1.4;3. The search for higher homotopy groupoids;119
9.1.5;4. Main results;125
9.1.6;5. Why crossed complexes?;127
9.1.7;6. Why cubical ?-groupoids with connections?;128
9.1.8;7. The equivalence of categories;129
9.1.9;8. First main aim of the work: higher Homotopy van Kampen theorems;130
9.1.10;9. The fundamental cubical ?-groupoid ?X* of a filtered space X*;131
9.1.11;10. Collapsing;133
9.1.12;11. Partial boxes;134
9.1.13;12. Thin elements;135
9.1.14;13. Sketch proof of the HHvKT;135
9.1.15;14. Tensor products and homotopies;137
9.1.16;15. Free crossed complexes and free crossed resolutions;139
9.1.17;16. Classifying spaces and the homotopy classification of maps;140
9.1.18;17. Relation with chain complexes with a groupoid of operators;141
9.1.19;18. Crossed complexes and simplicial groups and groupoids;143
9.1.20;19. Other homotopy multiple groupoids;144
9.1.21;20. Conclusion and questions;145
9.1.22;References;146
10;Section 4H: Hopf Algebras and Related Structures;152
10.1;Chapter 4. Hopf Algebraic Approach to Picard–Vessiot Theory;154
10.1.1;Introduction;155
10.1.2;Part I: PV theory in the differential context;157
10.1.3;Part II: PV theory in the C-ferential context;176
10.1.4;Part III: Unified PV theory;186
10.1.5;References;197
10.2;Chapter 5. Hopf Algebroids;200
10.2.1;1. Introduction;201
10.2.2;2. R-rings and R-corings;203
10.2.3;3. Bialgebroids;209
10.2.4;4. Hopf algebroids;231
10.2.5;Acknowledgment;259
10.2.6;References;259
10.3;Chapter 6. Comodules and Corings;264
10.3.1;1. Introduction;265
10.3.2;2. Categorical preliminaries;266
10.3.3;3. Corings;271
10.3.4;4. Comodules;281
10.3.5;5. Special types of corings and comodules;295
10.3.6;6. Applications;315
10.3.7;7. Extensions and dualizations;328
10.3.8;Acknowledgments;338
10.3.9;References;338
10.4;Chapter 7. Witt vectors. Part 1;346
10.4.1;1. Introduction and delimitation;352
10.4.2;2. Terminology;356
10.4.3;3. The p-adic Witt vectors. More historical motivation;356
10.4.4;4. Teichmüller representatives;358
10.4.5;5. Construction of the functor of the p-adicWitt vectors;358
10.4.6;6. The ring of p-adic Witt vectors over a perfect ring of characteristic p;365
10.4.7;7. Cyclic Galois extension of degree pn over a field of characteristic p;371
10.4.8;8. Cyclic central simple algebras of degree pn over a field of characteristic p;373
10.4.9;9. The functor of the big Witt vectors;375
10.4.10;10. The Hopf algebra Symm as the representing algebra for the bigWitt vectors;390
10.4.11;11. QSymm, the Hopf algebras of quasisymmetric functions and NSymm, the Hopf algebra of noncommutative symmetric functions;396
10.4.12;12. Free, cofree and duality properties of Symm;404
10.4.13;13. Frobenius and Verschiebung and other endomorphisms of A and the Witt vectors.;409
10.4.14;14. Supernatural and other quotients of the big Witt vectors;420
10.4.15;15. Cartier algebra and Dieudonné algebra;425
10.4.16;16. More operations on the and W functors: l-rings.;436
10.4.17;17. Necklace rings;454
10.4.18;18. Symm vs n R(Sn);461
10.4.19;19. Burnside rings;469
10.4.20;Appendix. The algebra of symmetric functions in in.nitely many indeterminates;479
10.4.21;References;481
10.5;Chapter 8. Crystal Graphs and the Combinatorics of Young Tableaux;500
10.5.1;1. Introduction;501
10.5.2;2. Quantum group and crystal base;501
10.5.3;3. Crystals and Young tableaux;507
10.5.4;4. Crystal equivalence;513
10.5.5;5. Bicrystals;518
10.5.6;References;530
11;Section 6D: Representation Theory of Algebras;532
11.1;Chapter 9. Quivers and Representations;534
11.1.1;1. Introduction;536
11.1.2;2. Kac–Moody algebras and quantum groups;541
11.1.3;3. Quivers and their representations;546
11.1.4;4. Dimension vectors and positive roots;560
11.1.5;5. Ringel–Hall algebras;563
11.1.6;6. PBW basis and canonical basis;566
11.1.7;7. Root categories, Kac–Moody algebras and elliptic Lie algebras;577
11.1.8;8. Guide to the literature;586
11.1.9;Acknowledgements;586
11.1.10;References;586
12;Section 8: Applied Algebra;590
12.1;Chapter 10. Canonical Decompositions and Invariants for Data Analysis;592
12.1.1;1. Introduction;593
12.1.2;2. A general principle of interpretability;593
12.1.3;3. Symmetry studies;594
12.1.4;4. The regular invariants of S3;598
12.1.5;5. The regular invariants of S4;603
12.1.6;6. Comments and summary;609
12.1.7;Appendix A;610
12.1.8;References;611
13;Index;612