Journal of Lie Theory 35 (2025), No. 4, 737--786
Copyright Heldermann Verlag 2025
On Group and Loop Spheres
Wolfgang Bertram
Institut Elie Cartan de Lorraine, Site de Nancy, Vandoeuvre, France
wolfgang.bertram@univ-lorraine.fr
[Abstract-pdf]
\newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bO}{\mathbb{O}} \newcommand{\K}{\mathbb{K}} \newcommand{\R}{\mathbb{R}} We investigate the problem of {\em 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 {\em composition algebras}; in particular, for $\K=\R$ and positive definite forms, we obtain the sequence of the four real division algebras $\R,\bC,\bH$ (quaternions), $\bO$ (octonions). Our theory is more general, allowing that $\K$ is merely a commutative 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 {\em ternary algebraic structures}, thus defining categories of {\em group spherical} and of {\em Moufang spherical spaces}. In particular, we develop a theory of {\em 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 {\em every $2$-dimensional quadratic space carries a canonical structure of commutative group spherical space}.
Keywords: Composition algebra, quaternions, octonions, (binary) quadratic form, sphere, generalized dicyclic group, circle group, group spherical space, Moufang loop spherical space, torsor, ternary loop.
MSC: 11E04, 11E16, 11H56, 11R11, 11R52, 15A63, 16S99, 17A40, 17A75, 17C37, 20N05, 20N10, 51N30.
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