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On group and loop spheres

Wolfgang Bertram

Abstract

We investigate the problem of defining group or loop structures on spheres, where by ''sphere'' we mean the level set q(x) = c of a general K-valued quadratic form q, for an invertible scalar c. When K is a field and q non-degenerate, then this corresponds to the classical theory of composition algebras; in particular, for K = R and positive definite forms, we obtain the sequence of the four real division algebras R, C, H (quaternions), O (octonions). Our theory is more general, allowing that K is merely a ring, and the form q possibly degenerate. To achieve this goal, we give a more geometric formulation, replacing the theory of binary composition algebras by ternary algebraic structures, thus defining categories of group spherical and of Moufang spherical spaces. In particular, we develop a theory of ternary Moufang loops, and show how it is related to the Albert-Cayley-Dickson construction and to generalized ternary octonion algebras. At the bottom, a starting point of the whole theory is the (elementary) result that every 2-dimensional quadratic space carries a canonical structure of commutative group spherical space.

On group and loop spheres

Abstract

We investigate the problem of defining group or loop structures on spheres, where by ''sphere'' we mean the level set q(x) = c of a general K-valued quadratic form q, for an invertible scalar c. When K is a field and q non-degenerate, then this corresponds to the classical theory of composition algebras; in particular, for K = R and positive definite forms, we obtain the sequence of the four real division algebras R, C, H (quaternions), O (octonions). Our theory is more general, allowing that K is merely a ring, and the form q possibly degenerate. To achieve this goal, we give a more geometric formulation, replacing the theory of binary composition algebras by ternary algebraic structures, thus defining categories of group spherical and of Moufang spherical spaces. In particular, we develop a theory of ternary Moufang loops, and show how it is related to the Albert-Cayley-Dickson construction and to generalized ternary octonion algebras. At the bottom, a starting point of the whole theory is the (elementary) result that every 2-dimensional quadratic space carries a canonical structure of commutative group spherical space.

Paper Structure

This paper contains 61 sections, 37 theorems, 147 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Spheres and groups.
  3. Binary versus ternary algebra.
  4. All circles are torsors.
  5. Group spherical spaces.
  6. Moufang spherical spaces.
  7. The Albert-Cayley-Dickson (ACD) construction.
  8. Further topics.
  9. All spheres are symmetric spaces
  10. Spheres.
  11. The Jordan structure.
  12. Existence of $q$-compatible ternary products.
  13. Reflection spaces and symmetric spaces.
  14. Spheres as symmetric spaces.
  15. The set of root vectors.
  16. ...and 46 more sections

Key Result

Lemma 2.2

For any quadratic space $(V,q)$ and $e \in V$, the operator $Q_e$ satisfies: It follows that, if $q(e)$ is invertible in $\mathbb{K}$, then $s_e:V \to V$, defines an isometry of order $2$, fixing $e$.

Figures (1)

  • Figure 1: Construction of the group law on the circle

Theorems & Definitions (134)

  • Remark 2.1
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.3
  • ...and 124 more