Journal of Lie Theory

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 $\mathbb{K}$-valued quadratic form $q$, for an invertible scalar $c$. When $\mathbb{K}$ is a field and $q$ non-degenerate, then this corresponds to the classical theory of composition algebras; in particular, for $\mathbb{K}=\mathbb{R}$ and positive definite forms, we obtain the sequence of the four real division algebras $\mathbb{R},\mathbb{C},\mathbb{H}$ (quaternions), $\mathbb{O}$ (octonions). Our theory is more general, allowing that $\mathbb{K}$ 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 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.