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The quaternion algebra is one of the most important and well-studied objects in mathematics and physics; and it has natural Hermitian form which induces Euclidean metric [1].
A functional ([dot], [dot]) : X x X [right arrow] K is said to be a Hermitian form on X if
We use the following notations related to a given Hermitian form ([dot], [dot]) on X:
The following result concerning the functional properties of [sigma] as a function depending on the nonnegative hermitian form ([dot], [dot]) has been obtained in [?
not equal to] 2, D a division k-algebra with center K = k([square root of (a)]) and involution J of second kind, non-trivial on K and h a non-degenerate hermitian form of rank n with respect to J and with values in D.
1, every hermitian form h of dimension n [greater than or equal to] 3 is isotropic over k.
A hermitian form over (D,[sigma]) is a pair (V,h) where V is a D-vector space and h is a map h : V x V [right arrow] D which is [sigma]-sesquilinear in the first argument, D-linear in the second argument and which satisfies
K] then a hermitian form is a symmetric bilinear form which can be identified with a quadratic form as char(K) [not equal to] 2.
His account relies on the basic results in the linear representations of finite groups in quadratic, symplectic, and Hermitian forms and in involutions over simple algebras, which are outlined, mainly without proofs, in the first chapter.
1)], which relates hermitian forms over a quaternion algebra with canonical involution--the unique symplectic involution on a quaternion algebra--to quadratic forms over the center.
There are many other well-known similar results (local global principles) in other contexts, say Hasse Minkowski Theorem for quadratic forms, Landherr Theorem for hermitian forms, etc.