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Collineations of Polar Spaces

Published online by Cambridge University Press:  20 November 2018

Donald G. James
Donald G. James*
Affiliation:
The Pennsylvania State University, University Park, Pennsylvania
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The fundamental theorem of projective geometry describes the bijective collineations between two projective spaces PV and PV′ of finite dimension (greater than one) over division rings k and k′ in terms of an isomorphism φ:k → k′ and a φ-semilinear bijective mapping between the underlying vector spaces V and V′. Tits [9, Theorem 8.611] has given an extensive generalization of this theorem to embeddable polar spaces induced by polarities coming from either (σ, )-hermitian forms or from (σ, )-quadratic forms with Witt indices at least two. In another direction, Klingenberg [7] and later André [1] and Rado [8], have generalized the fundamental theorem by considering non-injective collineations. Now the isomorphism φ must be replaced by a place φ:k → k′ ∪ ∞ and an integral structure over the valuation ring A = φminus1(k′) is induced into the projective space PV.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 2 , 01 April 1985 , pp. 296 - 309
Copyright
Copyright © Canadian Mathematical Society 1985

References

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