Vol. 15, No. 1, 2017

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Embedded polar spaces revisited

Antonio Pasini

Vol. 15 (2017), No. 1, 31–72

Pseudo-quadratic forms have been introduced by Tits in his Buildings of spherical type and finite BN-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 K and an admissible pair (σ,ε) in it, the codomain of a (σ,ε)-quadratic form is the group K¯ := KKσ,ε, where Kσ,ε := {t tσε}tK. Our generalization amounts to replace K¯ with a quotient K¯R¯ for a subgroup R¯ of K¯ such that λσR¯λ = R¯ for any λ K. We call generalized pseudo-quadratic forms (also generalized (σ,ε)-quadratic forms) the forms defined in this more general way, keeping the words pseudo-quadratic form and (σ,ε)-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 q : V K¯R¯ is a non-trivial generalized pseudo-quadratic form and f : V × V K is its sesquilinearization, the points and the lines of PG(V ) where q vanishes form a subspace Sq of the polar space Sf associated to f. 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 e : S PG(V ) be a projective embedding of a non-degenerate polar space S of rank at least 2; then e(S) is either the polar space Sq associated to a generalized pseudo-quadratic form q or the polar space Sf associated to an alternating form f. By exploiting this theorem we also obtain an elementary proof of the following well known fact: an embedding e as above is dominant if and only if either e(S) = Sq for a pseudo-quadratic form q or char(K)2 and e(S) = Sf for an alternating form f.

polar spaces, embeddings
Mathematical Subject Classification 2010
Primary: 51A45, 51A50, 51E12, 51E24
Received: 24 November 2014
Accepted: 8 August 2015
Antonio Pasini
Department of Information Engineering and Mathematics
University of Siena
Via Roma 56
53100 Siena