Global left loop structures on spheres

Kinyon, Michael K.

Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) no. 2, pp. 325-346

Résumé

On the unit sphere $\Bbb S$ in a real Hilbert space $\bold H$, we derive a binary operation $\odot $ such that $(\Bbb S,\odot )$ is a power-associative Kikkawa left loop with two-sided identity $\bold e_{0}$, i.e., it has the left inverse, automorphic inverse, and $A_l$ properties. The operation $\odot $ is compatible with the symmetric space structure of $\Bbb S$. $(\Bbb S,\odot )$ is not a loop, and the right translations which fail to be injective are easily characterized. $(\Bbb S,\odot )$ satisfies the left power alternative and left Bol identities ``almost everywhere'' but not everywhere. Left translations are everywhere analytic; right translations are analytic except at $-\bold e_{0}$ where they have a nonremovable discontinuity. The orthogonal group $O(\bold H)$ is a semidirect product of $(\Bbb S,\odot )$ with its automorphism group. The left loop structure of $(\Bbb S,\odot )$ gives some insight into spherical geometry.

Détail

Classification : 20N05, 58B25
Keywords: loop; quasigroup; sphere; Hilbert space; spherical geometry