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 connected compact Lie group of real dimension and is also called the unitary unimodular group. The determinant gives the exact sequence

(2)

Its Lie algebra consists of the antihermitian matrices having matrix trace zero,

(3)

where denotes the set of all complex matrices. The group center consists of the scalar matrices , where is an th root of unity and is the identity matrix. Factoring by the group center gives the projective special unitary group . The group is simply connected.

The notation instead denotes a finite special unitary group. Its elements are matrices over the finite field that preserve a nondegenerate Hermitian form and have determinant 1. For a finite group , denotes its group order. In particular,

(4)

The group can be represented by matrices

(5)

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

(6)

or the matrices

			(7)
			(8)
			(9)

The first parameterization 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

(10)

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

The group representation of dimension , whose entries are Wigner D-functions, is

(11)

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

			(12)
			(13)