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Global left loop structures on spheres

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Michael K. Kinyon

Global left loop structures on spheres

Comment.Math.Univ.Carolinae 41,2 (2000) 325-346.

Abstract: 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 identitye0, i.e., it has the left inverse, automorphic inverse, andAlproperties. The operation¯is compatible with the symmetric space structure ofS. (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 −e0 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.

Keywords: loop, quasigroup, sphere, Hilbert space, spherical geometry AMS Subject Classification: 20N05

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