On group and loop spheres

TL;DR

This paper explores the extension of classical sphere structures to more general quadratic forms over rings, developing a geometric and algebraic framework involving ternary structures and Moufang loops, broadening the scope of composition algebra theory.

Contribution

It introduces a generalized theory of group and loop structures on spheres defined by quadratic forms over rings, using ternary algebraic structures and Moufang loops.

Findings

Established a geometric formulation for spheres via quadratic forms.

Developed a theory of ternary Moufang loops and their relation to octonion algebras.

Proved that every 2-dimensional quadratic space has a canonical commutative group spherical structure.

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.