79 research outputs found
Global Left Loop Structures on Spheres
On the unit sphere in a real Hilbert space , we
derive a binary operation such that is a
power-associative Kikkawa left loop with two-sided identity ,
i.e., it has the left inverse, automorphic inverse, and properties. The
operation is compatible with the symmetric space structure of
. is not a loop, and the right translations
which fail to be injective are easily characterized.
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 where they have a
nonremovable discontinuity. The orthogonal group is a
semidirect product of with its automorphism group (cf.
http://www.arxiv.org/abs/math.GR/9907085). The left loop structure of
gives some insight into spherical geometry.Comment: 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 refere
Infinite Simple Bol Loops
If the left multiplication group of a loop is simple, then the loop is
simple. We use this observation to give examples of infinite simple Bol loops.Comment: 4 pages, AMS-LaTeX, to appear in Comment. Math. Univ. Carolinae for a
special issue: the Proceedings of Loops03. Version 3: more minor changes
suggested by the refere
Axioms for trimedial quasigroups
We give new equations that axiomatize the variety of trimedial quasigroups.
We also improve a standard characterization by showing that right semimedial,
left F-quasigroups are trimedial.Comment: 6 pages, AMS-LaTeX. To appear in Comment. Math. Univ. Carolinae. for
a special issue: the Proceedings of Loops03. Version 3: the proof of the main
result is collected together more formally; other stylistic change
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