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Loops in Group Theory and Lie Theory
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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