Innovations in Incidence Geometry
- Innov. Incidence Geom.
- Volume 15, Number 1 (2017), 31-72.
Embedded polar spaces revisited
Pseudo-quadratic forms have been introduced by Tits in his Buildings of spherical type and finite -pairs (1974), in view of the classification of polar spaces. A slightly different notion is proposed by Tits and Weiss. In this paper we propose a generalization. With its help we will be able to clarify a few points in the classification of embedded polar spaces. We recall that, according to Tits’ book, given a division ring and an admissible pair in it, the codomain of a -quadratic form is the group , where . Our generalization amounts to replace with a quotient for a subgroup of such that for any . We call generalized pseudo-quadratic forms (also generalized -quadratic forms) the forms defined in this more general way, keeping the words pseudo-quadratic form and -quadratic form for those defined as in Tits’ book. Generalized pseudo-quadratic forms behave just like pseudo-quadratic forms. In particular, every non-trivial generalized pseudo-quadratic form admits a unique sesquilinearization, characterized by the same property as the sesquilinearization of a pseudo-quadratic form. Moreover, if is a non-trivial generalized pseudo-quadratic form and is its sesquilinearization, the points and the lines of where vanishes form a subspace of the polar space associated to . In this paper, after a discussion of quotients and covers of generalized pseudo-quadratic forms, we shall prove the following, which sharpens a celebretated theorem of Buekenhout and Lefèvre. Let be a projective embedding of a non-degenerate polar space of rank at least ; then is either the polar space associated to a generalized pseudo-quadratic form or the polar space associated to an alternating form . By exploiting this theorem we also obtain an elementary proof of the following well known fact: an embedding as above is dominant if and only if either for a pseudo-quadratic form or and for an alternating form .
Innov. Incidence Geom., Volume 15, Number 1 (2017), 31-72.
Received: 24 November 2014
Accepted: 8 August 2015
First available in Project Euclid: 28 February 2019
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 51A45: Incidence structures imbeddable into projective geometries 51A50: Polar geometry, symplectic spaces, orthogonal spaces 51E12: Generalized quadrangles, generalized polygons 51E24: Buildings and the geometry of diagrams
Pasini, Antonio. Embedded polar spaces revisited. Innov. Incidence Geom. 15 (2017), no. 1, 31--72. doi:10.2140/iig.2017.15.31. https://projecteuclid.org/euclid.iig/1551323009