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Monatshefte für Mathematik

, Volume 187, Issue 3, pp 437–458 | Cite as

Clifford parallelisms defined by octonions

  • Andrea Blunck
  • Norbert Knarr
  • Bernhild Stroppel
  • Markus J. StroppelEmail author
Article

Abstract

We define (left and right) Clifford parallelisms on a seven-dimensional projective space algebraically, using an octonion division algebra. Thus, we generalize the two well-known Clifford parallelisms on a three-dimensional projective space, obtained from a quaternion division algebra. We determine (for both the octonion and quaternion case) the automorphism groups of these parallelisms. A geometric description of the parallel classes is given with the help of a hyperbolic quadric in a Baer superspace, obtained from the split octonion algebra over a quadratic extension of the ground field, again generalizing results that are known for the quaternion case. In contrast to the quaternion case, the orbits of the two Clifford parallelisms under the group of direct similitudes of the norm form of the algebra are non-trivial in the octonion case. The two spaces of parallelisms can be seen as the point sets of two point-line geometries, both isomorphic to the seven-dimensional projective space. Together with the original space, we thus have three versions of this projective space. We introduce a triality between them which is closely related to the triality of the polar space of split type \(\mathrm {D}_4\).

Keywords

Clifford parallelism Octonion Quaternion Composition algebra Projective space Triality 

Mathematics Subject Classification

51A15 17A75 51A05 51A10 51J15 11E04 

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Copyright information

© Springer-Verlag GmbH Austria 2017

Authors and Affiliations

  • Andrea Blunck
    • 1
  • Norbert Knarr
    • 2
  • Bernhild Stroppel
    • 2
  • Markus J. Stroppel
    • 2
    Email author
  1. 1.Department of MathematicsUniversität HamburgHamburgGermany
  2. 2.LExMathUniversität StuttgartStuttgartGermany

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