View source for Pseudo-quadratic form

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Let  $  D $
be a division ring (cf. [[Division algebra|Division algebra]]) with centre  $  k $
and let  $  V $
be a right vector space over  $  D $. 
Let  $  \sigma $
be an automorphism of  $  D $
and  $  \epsilon \in D $
such that  $  \epsilon \sigma ( \epsilon ) = 1 $, 
$  \sigma  ^ {2} ( x) = \epsilon x \epsilon  ^ {-} 1 $
for all  $  x \in D $. 
Assume also that  $  \epsilon \not\equiv - 1 $
if  $  \sigma =  \mathop{\rm id} $
and  $   \mathop{\rm char} ( D) \neq 2 $. 
Set

$$ 
D ( \sigma , \epsilon )  =  \{ {x - \sigma ( x) \epsilon } : {x \in D } \}
.
$$

This is an additive subgroup of  $  D $. 
Let  $  \overline{D}\; $
be the quotient  $  \overline{D}\; = D / D ( \sigma , \epsilon ) $, 
and let  $  x \mapsto \overline{x}\; $
denote the quotient mapping  $  D \rightarrow \overline{D}\; $. 
A pseudo-quadratic form on  $  V $
is a function  $  q :  V \rightarrow \overline{D}\; $
such that there exists a trace-valued  $  ( \sigma - \epsilon ) $-
Hermitian form (cf. [[Sesquilinear form|Sesquilinear form]])  $  f :  V \times V \rightarrow D $
such that  $  q ( v+ w) = q( v) + q( w) + \overline{ {f( v , w) }}\; $. 
The form  $  f $
is uniquely determined by this and is called the sesquilinearization of  $  q $.

A  $  (  \mathop{\rm id} , 1) $-
pseudo-quadratic form is a [[Quadratic form|quadratic form]] in the usual sense. The Witt index of a pseudo-quadratic form is that of the associated sesquilinear form.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Tits,   "Buildings and BN-pairs of spherical type" , Springer  (1974)  pp. Sect. 8.2</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Bourbaki,   "Eléments de mathématique. Algèbre" , Hermann  (1959)  pp. Chapt. 9. Formes sesquilinéaires et formes quadratiques</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.A. Dieudonné,   "La géométrie des groupes classiques" , Springer  (1963)</TD></TR>
</table>