Unitary Lie algebra - Knowlpedia

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: it splits as the direct sum of its center and the semisimple ideal $\mathfrak{su}(n)$ ; see compact Lie algebras are reductive.