Special unitary Lie algebra

Definition

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

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

Basic structure and context

• 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).