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

This is a real form (the group Sp(2) can be also defined as the subgroup of GL(2, H) which preserves the

Hermitian form (x, y) = [[summation].sup.2.sub.i=1] [[bar.x].sub.i][y.sub.i], where [x.sub.i], [y.sub.i], [member of] H and [[bar.x].sub.i] is the quaternionic conjugate of [x.sub.i]) of Sp(4, C), called Sp(2) (or sometimes USp(4)).

The Strominger system admits a conformally balanced

Hermitian form on a three-dimensional compact complex manifold M, a nowhere vanishing holomorphic (3,0)-form, and a HYM connection on a vector bundle E over this manifold.

[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.

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

In this section, we follow the exposition given in [18] in order to realize [N.sub.[omega]] := C [x.sub.[omega]] [C.sup.n] as a central extension of the Heisenberg group [H.sub.2n+1] := R [x.sub.Tm[omega]] [C.sup.n], where [omega](z, [omega]) denotes the standard

Hermitian form on [C.sup.n].

where g is a

Hermitian form of signature (2,2) and SU(2,2) is defined by

Ge, "The weight distribution of a class of cyclic codes related to

Hermitian forms graphs," IEEE Transactions on Information Theory, vol.

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.

10, (1.1)], 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.

SZEGO, On the eigenvalues of certain

Hermitian forms, Arch.