Pseudoquadratic 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
$\left(\sigma ,\epsilon \right)$ in it, the codomain
of a
$\left(\sigma ,\epsilon \right)$quadratic form
is the group
$\overline{K}:=K\u2215{K}_{\sigma ,\epsilon}$, where
${K}_{\sigma ,\epsilon}:={\left\{t{t}^{\sigma}\epsilon \right\}}_{t\in K}$. Our generalization
amounts to replace
$\overline{K}$
with a quotient
$\overline{K}\u2215\overline{R}$
for a subgroup
$\overline{R}$
of
$\overline{K}$ such
that
${\lambda}^{\sigma}\overline{R}\lambda =\overline{R}$ for
any
$\lambda \in K$.
We call
generalized pseudoquadratic forms (also
generalized
$\left(\sigma ,\epsilon \right)$quadratic
forms) the forms defined in this more general way, keeping the words
pseudoquadratic form
and
$\left(\sigma ,\epsilon \right)$quadratic
form for those defined as in Tits’ book. Generalized pseudoquadratic forms behave
just like pseudoquadratic forms. In particular, every nontrivial generalized
pseudoquadratic form admits a unique sesquilinearization, characterized by the same
property as the sesquilinearization of a pseudoquadratic form. Moreover, if
$q:V\to \overline{K}\u2215\overline{R}$
is a nontrivial generalized pseudoquadratic form and
$f:V\times V\to K$
is its sesquilinearization, the points and the lines of
$PG\left(V\right)$ where
$q$ vanishes form
a subspace
${S}_{q}$ of
the polar space
${S}_{f}$
associated to
$f$.
In this paper, after a discussion of quotients and covers of generalized pseudoquadratic forms,
we shall prove the following, which sharpens a celebretated theorem of Buekenhout and
Lefèvre. Let
$e:S\to PG\left(V\right)$
be a projective embedding of a nondegenerate polar space
$S$ of rank at
least
$2$; then
$e\left(S\right)$ is either the
polar space
${S}_{q}$
associated to a generalized pseudoquadratic form
$q$ or the polar space
${S}_{f}$ 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
$e\left(S\right)={S}_{q}$ for a
pseudoquadratic form
$q$
or
$char\left(K\right)\ne 2$ and
$e\left(S\right)={S}_{f}$ for an alternating
form
$f$.
