Pseudo-Quadratic Form

Let $ D $ be a division ring (cf. 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) $ 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 in the usual sense. The Witt index of a pseudo-quadratic form is that of the associated sesquilinear form.

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