Unitary group $U(n)$

Definition

The unitary group is

where $U^\ast=\overline{U}^{\,T}$. Equivalently, $U(n)$ is the group of complex-linear automorphisms of $\mathbb C^n$ preserving the standard Hermitian inner product $\langle v,w\rangle = v^\ast w$.

Since the defining equation $U^\ast U=I$ is closed, $U(n)$ is a closed subgroup of the general linear group; thus it is a Lie subgroup by the closed subgroup theorem. It is also compact.

Basic structure

The determinant map $\det:U(n)\to U(1)$ is a Lie group homomorphism with kernel the special unitary group $SU(n)$. For $n=1$, $U(1)$ is the circle group (see the $U(1)$ example).

Lie algebra

The Lie algebra of $U(n)$ is the unitary Lie algebra $\\mathfrak{u}(n)$, consisting of skew-Hermitian matrices, obtained by differentiating $U^\ast U=I$ at the identity (compare Lie algebras of Lie groups).

Context

$U(n)$ is the prototype compact matrix Lie group; averaging over $U(n)$ is a standard way to construct invariant inner products and prove complete reducibility results for representations (compare Weyl’s theorem).