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Format: 1 online resource (370p.)

Edition: Reprint 2011

ISBN: 9783110900583

Series Statement: De Gruyter Expositions in Mathematics ; 35

Content: In this book the theory of binary systems is considered as a part of group theory and, in particular, within the framework of Lie groups. The novelty is the consequent treatment of topological and differentiable loops as topological and differentiable sections in Lie groups. The interplay of methods and tools from group theory, differential geometry and topology, symmetric spaces, topological geometry, and the theory of foliations is what gives a special flavour to the results presented in this book. It is the first monograph devoted to the study of global loops. So far books on differentiable loops deal with local loops only. This theory can only be used partially for the theory of global loops since non-associative local structures have, in general, no global forms. The text is addressed to researchers in non-associative algebra and foundations of geometry. It should prove enlightening to a broad range of readers, including mathematicians working in group theory, the theory of Lie groups, in differential and topological geometry, and in algebraic groups. The authors have produced a text that is suitable not only for a graduate course, but also for selfstudy in the subjectby interested graduate students. Moreover, the material presented can be used for lectures and seminars in algebra, topological algebra and geometry.

Note: Frontmatter -- , Preface -- , Notation -- , Contents -- , Introduction -- , Part I. General theory of transitive sections in groups and the geometry of loops -- , Section 1. Elements of the theory of loops and the relations with group theory -- , Section 2. Scheerer extensions of loops -- , Section 3. Nets associated with loops -- , Section 4. Local 3-nets -- , Section 5. Loop-sections covered by 1-parameter subgroups and geodesic loops -- , Section 6. Bol loops and symmetric spaces -- , Section 7. Bol nets -- , Section 8. Strongly topological and analytic Bol loops -- , Section 9. Core of a Bol loop and Bruck loops -- , Section 10. Bruck loops and symmetric quasigroups over groups -- , Section 11. Topological and differentiable Bruck loops -- , Section 12. Bruck loops in algebraic groups -- , Section 13. Core-related Bol loops -- , Section 14. Products and loops as sections in compact Lie groups -- , Section 15. Loops on symmetric spaces of groups -- , Section 16. Loops with compact translation groups and compact Bol loops -- , Section 17. Loops having a sharply transitive normal subgroup in their group generated by the left translations -- , Part II. Smooth loops on low dimensional manifolds -- , Section 18. Loops on 1-manifolds having a locally compact left translation group -- , Section 19. Topological loops on 2-dimensional manifolds -- , Section 20. Topological loops on tori -- , Section 21. Topological loops on the cylinder and on the plane -- , Section 22. The hyperbolic plane loop and its isotopism class -- , Section 23. Two-dimensional loops having a 3-dimensional solvable Lie group as their group generated by the left translations -- , Section 24. 2-dimensional Bol loops having a 4-dimensional Lie group as the group generated by their left translations -- , Section 25. Classification of differentiable 2-dimensional Bol loops -- , Section 26. Collineation groups of 4-dimensional Bol nets -- , Section 27. Strongly left alternative plane left A-loops -- , Section 28. 2-dimensional loops having a Lie group as the group generated by all translations -- , Section 29. Multiplicative loops of locally compact connected quasifields -- , Bibliography -- , Index , In English.

Additional Edition: ISBN 978-3-11-017010-8

Language: English

DOI: 10.1515/9783110900583

URL: https://doi.org/10.1515/9783110900583

URL: https://doi.org/10.1515/9783110900583