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Clifford parallelisms defined by octonions. (English) Zbl 1400.51002

Summary: 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 \(\mathrm{D}_4\).

MSC:

51A15 Linear incidence geometric structures with parallelism
17A75 Composition algebras
51A05 General theory of linear incidence geometry and projective geometries
51A10 Homomorphism, automorphism and dualities in linear incidence geometry
51J15 Kinematic spaces
11E04 Quadratic forms over general fields
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