# 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. 1.

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

2. 2.

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

3. 3.

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

4. 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 HermitianFormOverADivisionRing 2013-03-22 15:41:04 2013-03-22 15:41:04 CWoo (3771) CWoo (3771) 12 CWoo (3771) Definition msc 15A63 Hermitian form skew Hermitian form