Special unitary group

The compact matrix Lie group SU(n) preserving a Hermitian form with determinant 1.

Special unitary group

Let $\langle\cdot,\cdot\rangle$ be the standard Hermitian inner product on $\mathbb C^n$ . The special unitary group is

SU(n)=\{A\in GL(n,\mathbb C): A^*A=I,\ \det(A)=1\},

a closed Lie subgroup of the unitary group , hence a Lie group (see closed subgroup ). The group $SU(n)$ is compact and connected, and for $n\ge 2$ it is simply connected (see simply connected Lie group ).

\mathfrak{su}(n)=\{X\in M_n(\mathbb C): X^*+X=0,\ \mathrm{tr}(X)=0\},

with bracket $[X,Y]=XY-YX$ . Because $SU(n)$ is compact, many structural results apply cleanly, including existence of bi-invariant metrics (compare compact implies bi-invariant metric ) and strong harmonic analysis statements such as the Peter–Weyl theorem and Schur orthogonality .

A notable low-rank case is $SU(2)$ (see SU(2) example ), which is isomorphic to the Spin(3) double cover of $SO(3)$ (see SO(3) example ).