| Title:
|
Global left loop structures on spheres (English) |
| Author:
|
Kinyon, Michael K. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
41 |
| Issue:
|
2 |
| Year:
|
2000 |
| Pages:
|
325-346 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
loop |
| Keyword:
|
quasigroup |
| Keyword:
|
sphere |
| Keyword:
|
Hilbert space |
| Keyword:
|
spherical geometry |
| MSC:
|
20N05 |
| MSC:
|
58B25 |
| idZBL:
|
Zbl 1041.20044 |
| idMR:
|
MR1780875 |
| . |
| Date available:
|
2009-01-08T19:01:44Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119167 |
| . |
| Reference:
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