Theory of K-Loops

Hubert Kiechle3540432620, 9783540432623

The book contains the first systematic exposition of the current known theory of K-loops, as well as some new material. In particular, big classes of examples are constructed. The theory for sharply 2-transitive groups is generalized to the theory of Frobenius groups with many involutions. A detailed discussion of the relativistic velocity addition based on the author’s construction of K-loops from classical groups is also included. The first chapters of the book can be used as a text, the later chapters are research notes, and only partially suitable for the classroom. The style is concise, but complete proofs are given. The prerequisites are a basic knowledge of algebra such as groups, fields, and vector spaces with forms.

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Table of contents :
41uOfpdQgQL……Page 1
front-matter……Page 2
Introduction……Page 12
A. Groups……Page 17
B. Permutation Groups……Page 18
C. Geometry……Page 20
D. Binomial Coefficients……Page 21
E. Ordered Fields……Page 22
F. Hermitian Matrices and the Polar Decomposition……Page 23
G. Miscellaneous Results for Matrices……Page 28
2. Left Loops and Transversals……Page 33
A. Left Loops and the Left Multiplication Group……Page 35
B. Transversals and Sections……Page 37
C. Transassociants and the Quasidirect Product……Page 48
A. The Left Inverse Property……Page 53
B. Kikkawa Left Loops……Page 57
C. The Bol Condition……Page 59
4. Isotopy Theory……Page 63
5. Nuclei and the Autotopism Group……Page 69
A. Left Power Alternative Left Loops……Page 75
B. Bol Loops……Page 78
C. K-Loops……Page 82
D. Half Embedding……Page 87
A. Reflection Structures……Page 92
B. Frobenius Groups……Page 94
C. Involutions……Page 98
D. Sharply 2-transitive groups……Page 100
E. Characteristic 2……Page 103
F. Characteristic not 2 and Specific Groups……Page 105
8. Loops with Fibrations……Page 112
A. The Construction……Page 116
B. General and Special Linear Groups……Page 118
C. Linear Groups over the Quaternions……Page 127
D. Symplectic Groups……Page 129
E. Pseudo-orthogonal and Pseudo-unitary Groups……Page 130
F. Fibrations……Page 141
A. Lorentz Boosts……Page 146
B. Special Relativity……Page 149
11. K-loops from the General Linear Groups over Rings……Page 152
A. General Theory……Page 160
B. eta-Derivations……Page 163
C. Examples……Page 167
13. Appendix……Page 174
14. References……Page 180
15 .Index……Page 190