SU(2)

In mathematics, the special unitary group of degree $n$ , denoted $SU(n)$ , is the Lie group of $n \times n$ unitary matrices with determinant 1. The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group $U(n)$ , consisting of all $n \times n$ unitary matrices. As a compact classical group, $U(n)$ is the group that preserves the standard inner product on $C n$ . It is itself a subgroup of the general linear group, $SU(n) \subset U(n) \subset GL(n, C)$ .

The simplest case, $SU(1)$ , is the trivial group, having only a single element. The group $SU(2)$ is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from $SU(2)$ to the rotation group $SO(3)$ whose kernel is ${+ I, - I$ }. $SU(2)$ is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations.