sesquilinear forms over general fields

Let $V$ be a vector space over a field $k$. $k$ may be of any characteristic.

1 Sesquilinear Forms

Definition 1.

A function $b:V\times V\rightarrow k$ is sesquilinear if it satisfies each of the following:

1. 1.

$b(v,w+u)=b(v,w)+b(v,u)$ and $b(v+u,w)=b(v,w)+b(u,w)$ for all $u,v,w\in V$;

2. 2.

For a given field automorphism $\theta$ of $k$, $b(v,lw)=l^{\theta}b(v,w)$ and $b(lv,w)=lb(v,w)$ for all $v,w\in V$ and $l\in k$.

Remark 2.

It is possible to apply the field automorphism in the first variable but is more common to do so in the second variable. Also, if $\theta=1$ the form is a bilinear form.

Sesquilinear forms are commonly ascribed any combination of the following properties:

• reflexive, (commonly required to define perpendicular);

• positive definite (this condition requires the fixed field of $\theta$, $k_{0}$, be an ordered field, such as the rationals $\mathbb{Q}$ or reals $\mathbb{R}$).

Non-degenerate sesquilinear and bilinear forms apply to projective geometries as dualities and polarities through the induced $\perp$ operation. (See polarity (http://planetmath.org/Polarity2).)

2 Hermitian Forms

If $\theta^{2}=1$, it is common to exchange notation at this point and use the same notation of $\bar{l}$ for $l^{\theta}$ as is common for complex conjugation – even if $k$ is not $\mathbb{C}$. Then $\bar{\bar{l}}=l$.

In this notation, Hermitian forms may be defined by the property

 $b(v,w)=\overline{b(w,v)}.$
Remark 3.

It is not uncommon to see hermitian or Hermitean instead of Hermitian. The name is a tribute to Charles Hermite of the Ecole Polytechnique.

 Title sesquilinear forms over general fields Canonical name SesquilinearFormsOverGeneralFields Date of creation 2013-03-22 15:58:17 Last modified on 2013-03-22 15:58:17 Owner Algeboy (12884) Last modified by Algeboy (12884) Numerical id 11 Author Algeboy (12884) Entry type Definition Classification msc 47A07 Classification msc 15A63 Classification msc 11E39 Classification msc 51A05 Synonym Hermitian form Synonym Hermitean form Related topic ReflexiveNonDegenerateSesquilinear Related topic NonDegenerate Related topic Polarity2 Related topic Projectivity Related topic ProjectiveGeometry Related topic Isometry2 Related topic ProjectiveGeometry3 Related topic ClassicalGroups Defines sesquilinear form Defines Hermitian form Defines bilinear form Defines Hermitean