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Buildings of type Cn. III. Non-embeddable polar spaces

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Part of the Lecture Notes in Mathematics book series (LNM,volume 386)

Abstract

THEOREM. Let K be a division Cayley algebra over a field k, and let nK : K → k be the quadratic norm form of this algebra. Then, there exists one and, up to isomorphism, only one polar space of rank 3 whose planes are projective planes over K. The cones of this space (7.2.6) are isomorphic to the dual of the polar space associated with the quadratic form K × k4 → k defined by

$$ (x_0 ,x_1 ,x_2 ,x_3 ,x_4 ) \mapsto n_K (x_0 ) - x_1 x_3 + x_2 x_4 , $$
(1)

where x0 ∈ K and xi ∈ k for i = 1, 2, 3, 4.

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© 1974 Springer-Verlag Berlin Heidelberg

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(1974). Buildings of type Cn. III. Non-embeddable polar spaces. In: Buildings of Spherical Type and Finite BN-Pairs. Lecture Notes in Mathematics, vol 386. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-38349-9_9

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  • DOI: https://doi.org/10.1007/978-3-540-38349-9_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-06757-3

  • Online ISBN: 978-3-540-38349-9

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