Unitary Lie algebra

The Lie algebra of $U(n)$: skew-Hermitian matrices with the commutator bracket.

Unitary Lie algebra

Definition

The unitary Lie algebra $\mathfrak{u}(n)$ is the Lie algebra of the unitary group $U(n)$ . Concretely,

\mathfrak{u}(n)=\{X\in M_n(\mathbb C)\mid X^\ast+X=0\},

with Lie bracket $[X,Y]=XY-YX$ .

It is a real Lie algebra of dimension $\dim_\mathbb{R}\mathfrak{u}(n)=n^2$ .

Center and derived subalgebra

The center is

Z(\mathfrak{u}(n))=\{i t\,I_n\mid t\in\mathbb R\},

since scalar matrices commute with everything, and skew-Hermitian forces the scalar to be purely imaginary (compare the center of a Lie algebra ). The derived subalgebra satisfies

[\mathfrak{u}(n),\mathfrak{u}(n)]=\mathfrak{su}(n),

where $\mathfrak{su}(n)$ is the special unitary Lie algebra .

Context

As the Lie algebra of a compact Lie group, $\mathfrak{u}(n)$ is reductive in the sense of compact Lie algebras are reductive : it splits as a direct sum of its center and a semisimple ideal (here $\mathfrak{su}(n)$ ). This decomposition is ubiquitous in unitary representation theory.