Pseudo-quadratic form

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)