Special unitary Lie algebra

The special unitary Lie algebra $\mathfrak{su}(n)$ is the real Lie algebra of the special unitary group $SU(n)$ . Concretely,

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

where $X^\ast=\overline{X}^{\,T}$ is the Hermitian adjoint. The Lie bracket is

[X,Y]=XY-YX.

Basic structure and context

As a real vector space, $\dim_\mathbb{R}\mathfrak{su}(n)=n^2-1$ .
Its center is trivial. In particular, for $n\ge2$ , $\mathfrak{su}(n)$ is a real simple Lie algebra.
The inclusion $\mathfrak{su}(n)\subset\mathfrak{gl}(n,\mathbb C)$ is the differential at the identity of $SU(n)\subset GL(n,\mathbb C)$ .

$\mathfrak{su}(n)$ is the compact real form of $\mathfrak{sl}(n,\mathbb C)$ , and its representation theory is a cornerstone of highest-weight methods.

Equivalent characterizations

Equivalently, $\mathfrak{su}(n)$ is the trace-zero, codimension-one ideal in the unitary Lie algebra $\mathfrak u(n)$ .