Unitary Lie algebra

Definition

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

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

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

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.