Special Unitary Group

The special unitary group is the subgroup of the unitary group consisting of unitary matrices with determinant 1,

(1)

It is a compact connected Lie group of real dimension and is also called the unitary unimodular group. The notation instead denotes a finite special unitary group: matrices over the finite field that preserve a nondegenerate Hermitian form and have determinant 1.

The group can be represented by matrices

(2)

where and are the Cayley-Klein parameters. The group may also be represented by matrices

(3)

or the matrices

			(4)
			(5)
			(6)

The first parametrization identifies with the group of unit quaternions, and hence with the 3-sphere . Conjugation of pure imaginary quaternions gives a surjective group homomorphism from onto the rotation group with kernel , so

(7)

Thus is simply connected and is the universal cover and double cover of .

The order representation, whose matrix elements are Wigner D-functions, is

(8)

The summation is terminated by putting . The group character is given by

			(9)
			(10)