Generalized Special Unitary Group
For a field F, the generalized special unitary group over F, SU(p, q; F), is the group of all linear transformations of determinant 1 of a vector space of rank n = p + q over F which leave invariant a nondegenerate, Hermitian form of signature (p, q). This group is often referred to as the special unitary group of signature p q over F. The field F can be replaced by a commutative ring, in which case the vector space is replaced by a free module.
Specifically, fix a Hermitian matrix A of signature p q in GL(n, R), then all
satisfy
Often one will see the notation SU(p, q) without reference to a ring or field, in this case the ring or field being referred to is C and this gives one of the classical Lie groups. The standard choice for A when F = C is
However there may be better choices for A for certain dimensions which exhibit more behaviour under restriction to subrings of C.
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