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# Hermitian form over a division ring

Let $D$ be a division ring admitting an involution $*$. 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. 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.

Defines:

Hermitian form, skew Hermitian form

Type of Math Object:

Definition

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Reference

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## Mathematics Subject Classification

15A63*no label found*

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