Loops in Group Theory and Lie Theory

Péter Nagy, Karl Strambach

• Publisher: De Gruyter
• Year: 2011
• Print ISBN: 9783110170108
• eBook ISBN: 9783110900583
• Edition: 1
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

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.

Table of contents

• Preface
• Notation
• Introduction
• Part I. General theory of transitive sections in groups and the geometry of loops
• 1 Elements of the theory of loops
• 1.1 Basic facts on loops
• 1.2 Loops as sections in groups
• 1.3 Topological loops and differentiable loops
• 2 Scheerer extensions of loops
• 3 Nets associated with loops
• 4 Local 3-nets
• 5 Loop-sections covered by 1-parameter subgroups and geodesic loops
• 6 Bol loops and symmetric spaces
• 7 Bol nets
• 8 Strongly topological and analytic Bol loops
• 9 Core of a Bol loop and Bruck loops
• 9.1 Core of a Bol loop
• 9.2 Symmetric spaces on differentiable Bol loops
• 10 Bruck loops and symmetric quasigroups over groups
• 11 Topological and differentiable Bruck loops
• 12 Bruck loops in algebraic groups
• 13 Core-related Bol loops
• 14 Products and loops as sections in compact Lie groups
• 14.1 Pseudo-direct products
• 14.2 Crossed direct products
• 14.3 Non-classical differentiable sections in compact Lie groups
• 14.4 Differentiable local Bol loops as local sections in compact Lie groups
• 15 Loops on symmetric spaces of groups
• 15.1 Basic constructions
• 15.2 A fundamental reduction
• 15.3 Core loops of direct products of groups
• 15.4 Scheerer extensions of groups by core loops
• 16 Loops with compact translation groups and compact Bol loops
• 17 Sharply transitive normal subgroups
• Part II. Smooth loops on low dimensional manifolds
• 18 Loops on 1-manifolds
• 19 Topological loops on 2-dimensional manifolds
• 20 Topological loops on tori
• 21 Topological loops on the cylinder and on the plane
• 21.1 2-dimensional topological loops on the cylinder
• 21.2 Non-solvable left translation groups
• 22 The hyperbolic plane loop and its isotopism class
• 23 3-dimensional solvable left translation groups
• 23.1 The loops L(α) and their automorphism groups
• 23.2 Sharply transitive sections in £2 × ℝ
• 23.3 Sections in the 3-dimensional non-abelian nilpotent Lie group
• 23.4 Non-existence of strongly left alternative loops
• 24 4-dimensional left translation group
• 25 Classification of differentiable 2-dimensional Bol loops
• 26 Collineation groups of 4-dimensional Bol nets
• 27 Strongly left alternative plane left A-loops
• 28 Loops with Lie group of all translations
• 29 Multiplicative loops of locally compact connected quasifields
• 29.1 2-dimensional locally compact quasifields
• 29.2 Rees algebras Qε
• 29.3 Mutations of classical compact Moufang loops
• Bibliography
• Index