Special unitary group SU(n)

Example

Let $n \ge 1$. The special unitary group is $$SU(n) = \set{U \in M_n(\C) : U^* U = I_n, \ \det U = 1},$$ where $U^*$ is the conjugate transpose of $U$ and $I_n$ is the identity matrix. It is the kernel of the determinant $\det \colon U(n) \to U(1)$ on the unitary group, so it is a normal subgroup of $U(n)$.

Check that it is a compact Lie group. The two conditions are polynomial equations in the real and imaginary parts of the entries, so $SU(n)$ is closed in $GL_n(\C)$. By the closed subgroup theorem it is a Lie group. Each column of a unitary matrix is a unit vector, so $SU(n)$ is bounded in $M_n(\C)$, hence compact.