Unitary group $U(n)$

The compact Lie group of complex matrices preserving the standard Hermitian inner product.

Unitary group $U(n)$

Definition

The unitary group is

U(n)=\{U\in GL(n,\mathbb C)\mid U^\ast U=I\},

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 ).