Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Embedded polar spaces revisited   In this paper we introduce generalized pseudo-quadratic forms and develope
some theory for them. Recall that the codomain of a
$(\sigma,\varepsilon)$-quadratic form is the group $\overline{K} :=
K/K_{\sigma,\varepsilon}$, where $K$ is the underlying division ring of the
vector space on which the form is defined and $K_{\sigma,\varepsilon} :=
\{t-t^\sigma\varepsilon\}_{t\in K}$. Generalized pseudo-quadratic forms are
defined in the same way as $(\sigma,\varepsilon)$-quadratic forms but for
replacing $\overline{K}$ with a quotient $\overline{K}/\overline{R}$ for a
subgroup $\overline{R}$ of $\overline{K}$ such that
$\lambda^\sigma\overline{R}\lambda = \overline{R}$ for any $\lambda\in K$. 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 $q:V\rightarrow
\overline{K}/\overline{R}$ is a non-trivial generalized pseudo-quadratic form
and $f:V\times V\rightarrow K$ is its sesquilinarization, the points and the
lines of $\mathrm{PG}(V)$ where $q$ vanishes form a subspace $S_q$ of the polar
space $S_f$ associated to $f$. After a discussion of quotients and covers of
generalized pseudo-quadratic forms we prove the following: let $e:S\rightarrow
\mathrm{PG}(V)$ be a projective embedding of a non-degenerate polar space $S$
of rank at least $2$; then $e(S)$ is either the polar space $S_q$ associated to
a generalized pseudo-quadratic 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 $e(S) = S_q$ for a pseudo-quadratic form $q$ or
$\mathrm{char}(K)\neq 2$ and $e(S) = S_f$ for an alternating form $f$.

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The categories of this paper are: hep-lat
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The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat
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The categories of this paper are: hep-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: funct-an
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: q-bio
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The categories of this paper are: cmp-lg
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The categories of this paper are: mtrl-th