Global Left Loop Structures on Spheres
| dc.creator | Kinyon, Michael K. | |
| dc.date | 1999-10-21 | |
| dc.date | 2000-02-10 | |
| dc.date.accessioned | 2026-07-07T05:31:14Z | |
| dc.date.available | 2026-07-07T05:31:14Z | |
| dc.description | On the unit sphere $\mathbb{S}$ in a real Hilbert space $\mathbf{H}$, we derive a binary operation $\odot$ such that $(\mathbb{S},\odot)$ is a power-associative Kikkawa left loop with two-sided identity $\mathbf{e}_0$, i.e., it has the left inverse, automorphic inverse, and $A_l$ properties. The operation $\odot$ is compatible with the symmetric space structure of $\mathbb{S}$. $(\mathbb{S},\odot)$ is not a loop, and the right translations which fail to be injective are easily characterized. $(\mathbb{S},\odot)$ 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 $-\mathbf{e}_0$ where they have a nonremovable discontinuity. The orthogonal group $O(\mathbf{H})$ is a semidirect product of $(\mathbb{S},\odot)$ with its automorphism group (cf. http://www.arxiv.org/abs/math.GR/9907085). The left loop structure of $(\mathbb{S},\odot)$ gives some insight into spherical geometry. | |
| dc.description | 18 pages, no figures, 10pt, LaTeX2e, uses amsart.cls & tcilatex.tex. To appear in Comment. Math. Univ. Carolin. (special issue: Proceedings of LOOPS99) Revised version: various fixes and improvements suggested by referee | |
| dc.identifier | https://arxiv.org/abs/math/9910111 | |
| dc.identifier | http://arxiv.org/abs/math/9910111 | |
| dc.identifier | Comment. Math. Univ. Carolinae 41 (2000), no. 2, 325-346 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/79268 | |
| dc.subject | Group Theory | |
| dc.subject | Differential Geometry | |
| dc.subject | 20N05 | |
| dc.title | Global Left Loop Structures on Spheres | |
| dc.type | text |