Pseudo-quadratic form
From Encyclopedia of Mathematics
Let 
be a division ring (cf. Division algebra) with centre 
and let 
be a right vector space over 
. Let 
be an automorphism of 
and 
such that 
, 
for all 
. Assume also that 
if 
and 
. Set
 |
This is an additive subgroup of 
. Let 
be the quotient 
, and let 
denote the quotient mapping 
. A pseudo-quadratic form on 
is a function 
such that there exists a trace-valued 
-Hermitian form (cf. Sesquilinear form) 
such that 
. The form 
is uniquely determined by this and is called the sesquilinearization of 
.
A 
-pseudo-quadratic form is a quadratic form in the usual sense. The Witt index of a pseudo-quadratic form is that of the associated sesquilinear form.
References
| [a1] | J. Tits, "Buildings and BN-pairs of spherical type" , Springer (1974) pp. Sect. 8.2 |
| [a2] | N. Bourbaki, "Eléments de mathématique. Algèbre" , Hermann (1959) pp. Chapt. 9. Formes sesquilinéaires et formes quadratiques |
| [a3] | J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1963) |
How to Cite This Entry:
Pseudo-quadratic form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-quadratic_form&oldid=48351
Pseudo-quadratic form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-quadratic_form&oldid=48351