Hermitian form over a division ring
Let be a division ring admitting an involution (http://planetmath.org/Involution2) . Let be a vector space over . A Hermitian form over is a function from to , denoted by with the following properties, for any and :
-
1.
is additive in each of its arguments,
-
2.
,
-
3.
,
-
4.
.
Note that if the Hermitian form is non-trivial and if is the identity on , then is a field and is just a symmetric bilinear form.
If we replace the last condition by , then over 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 |