Hermitian form over a division ring

Let $D$ be a division ring admitting an involution (http://planetmath.org/Involution2) $*$. Let $V$ be a vector space over $D$. A Hermitian form over $D$ is a function from $V\times V$ to $D$, denoted by $(\cdot ,\cdot )$ with the following properties, for any $v,w\in V$ and $d\in D$:

1. $(\cdot ,\cdot )$ is additive in each of its arguments,

2. $(du,v)=d(u,v)$,

3. $(u,dv)=(u,v)d^{*}$,

4. $(u,v)={(v,u)}^{*}$.

Note that if the Hermitian form $(\cdot ,\cdot )$ is non-trivial and if $*$ is the identity on $D$, then $D$ is a field and $(\cdot ,\cdot )$ is just a symmetric bilinear form.

If we replace the last condition by $(u,v)=-{(v,u)}^{*}$, then $(\cdot ,\cdot )$ over $D$ is called a skew Hermitian form.

Remark. Every skew Hermitian form over a division ring induces a Hermitian form and vice versa.

Title	Hermitian form over a division ring
Canonical name	HermitianFormOverADivisionRing
Date of creation	2013-03-22 15:41:04
Last modified on	2013-03-22 15:41:04
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	12
Author	CWoo (3771)
Entry type	Definition
Classification	msc 15A63
Defines	Hermitian form
Defines	skew Hermitian form