2024/10/23 by Wolfgang Bertram, Bertram, Wolfgang
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #FOS: Mathematics #Group Theory (math.GR) #History and Theory of Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2410.17634
openalex publication_date 2024/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.