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- W1857401500 abstract "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." @default.
- W1857401500 created "2016-06-24" @default.
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- W1857401500 date "2000-01-01" @default.
- W1857401500 modified "2023-09-24" @default.
- W1857401500 title "Global left loop structures on spheres" @default.
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