Special unitary Lie algebra

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

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 the matrix commutator

[X,Y]=XY-YX.

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

As a real vector space, $\dim_\mathbb{R}\mathfrak{su}(n)=n^2-1$ .
The center of $\mathfrak{su}(n)$ is trivial: if $X$ commutes with all of $\mathfrak{su}(n)$ , then $X$ is a scalar matrix, and tracelessness forces $X=0$ . In particular, for $n\ge 2$ , $\mathfrak{su}(n)$ is a (real) simple Lie algebra .
The inclusion $\mathfrak{su}(n)\subset \mathfrak{gl}(n,\mathbb C)$ is the differential at the identity of the defining inclusion $SU(n)\subset GL(n,\mathbb C)$ , as in the Lie algebra of a Lie group and the principle that the differential of a Lie group homomorphism is a Lie algebra homomorphism .

A standard motivation is that $\mathfrak{su}(n)$ is the compact real form of $\mathfrak{sl}(n,\mathbb C)$ (see $\\mathfrak{sl}(n,\\mathbb C)$ ), and its representation theory is a cornerstone of highest-weight methods (compare highest weights and the highest-weight theorem ).