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Mathematics 2015, 3(4), 1192-1221; doi:10.3390/math3041192

From Cayley-Dickson Algebras to Combinatorial Grassmannians

1
Astronomical Institute, Slovak Academy of Sciences, Tatranská Lomnica 05960, Slovak Republic
2
IRTES/UTBM, Université de Bourgogne Franche-Comté, Belfort Cedex 90010, France
3
J. Heyrovský Institute of Physical Chemistry, v. v. i., Academy of Sciences of the Czech Republic, Dolejškova 3, Prague 18223, Czech Republic
*
Author to whom correspondence should be addressed.
Academic Editor: Carsten Schneider
Received: 23 August 2015 / Revised: 26 November 2015 / Accepted: 30 November 2015 / Published: 4 December 2015

Abstract

Given a 2N -dimensional Cayley-Dickson algebra, where 3 ≤ N ≤ 6 , we first observe that the multiplication table of its imaginary units ea , 1 ≤ a ≤ 2N - 1 , is encoded in the properties of the projective space PG(N - 1,2) if these imaginary units are regarded as points and distinguished triads of them {ea, eb , ec} , 1 ≤ a < b < c ≤ 2N - 1 and eaeb = ±ec , as lines. This projective space is seen to feature two distinct kinds of lines according as a + b = c or a + b ≠ c . Consequently, it also exhibits (at least two) different types of points in dependence on how many lines of either kind pass through each of them. In order to account for such partition of the PG(N - 1,2) , the concept of Veldkamp space of a finite point-line incidence structure is employed. The corresponding point-line incidence structure is found to be a specific binomial configuration CN; in particular, C3 (octonions) is isomorphic to the Pasch (62, 43) -configuration, C4 (sedenions) is the famous Desargues (103) -configuration, C5 (32-nions) coincides with the Cayley-Salmon (154, 203) -configuration found in the well-known Pascal mystic hexagram and C6 (64-nions) is identical with a particular (215, 353) -configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration. Finally, a brief examination of the structure of generic CN leads to a conjecture that CN is isomorphic to a combinatorial Grassmannian of type G2(N + 1). View Full-Text
Keywords: Cayley-Dickson algebras; Veldkamp spaces; finite geometries Cayley-Dickson algebras; Veldkamp spaces; finite geometries
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Saniga, M.; Holweck, F.; Pracna, P. From Cayley-Dickson Algebras to Combinatorial Grassmannians. Mathematics 2015, 3, 1192-1221.

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