Global left loop structures on spheres

Michael K. Kinyon

Commentationes Mathematicae Universitatis Carolinae (2000)

  • Volume: 41, Issue: 2, page 325-346
  • ISSN: 0010-2628

Abstract

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On the unit sphere 𝕊 in a real Hilbert space 𝐇 , we derive a binary operation ⊙ such that ( 𝕊 , ⊙ ) is a power-associative Kikkawa left loop with two-sided identity 𝐞 0 , i.e., it has the left inverse, automorphic inverse, and A l properties. The operation ⊙ is compatible with the symmetric space structure of 𝕊 . ( 𝕊 , ⊙ ) is not a loop, and the right translations which fail to be injective are easily characterized. ( 𝕊 , ⊙ ) 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 - 𝐞 0 where they have a nonremovable discontinuity. The orthogonal group O ( 𝐇 ) is a semidirect product of ( 𝕊 , ⊙ ) with its automorphism group. The left loop structure of ( 𝕊 , ⊙ ) gives some insight into spherical geometry.

How to cite

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Kinyon, Michael K.. "Global left loop structures on spheres." Commentationes Mathematicae Universitatis Carolinae 41.2 (2000): 325-346. <http://eudml.org/doc/248593>.

@article{Kinyon2000,
abstract = {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. The left loop structure of $(\mathbb \{S\},\odot )$ gives some insight into spherical geometry.},
author = {Kinyon, Michael K.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {loop; quasigroup; sphere; Hilbert space; spherical geometry; quasigroups; spheres; Hilbert spaces; spherical geometry; Bol loops; Bruck loops; Kikkawa left loops; reflections},
language = {eng},
number = {2},
pages = {325-346},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Global left loop structures on spheres},
url = {http://eudml.org/doc/248593},
volume = {41},
year = {2000},
}

TY - JOUR
AU - Kinyon, Michael K.
TI - Global left loop structures on spheres
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 2
SP - 325
EP - 346
AB - 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. The left loop structure of $(\mathbb {S},\odot )$ gives some insight into spherical geometry.
LA - eng
KW - loop; quasigroup; sphere; Hilbert space; spherical geometry; quasigroups; spheres; Hilbert spaces; spherical geometry; Bol loops; Bruck loops; Kikkawa left loops; reflections
UR - http://eudml.org/doc/248593
ER -

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