sl(1,C) — complex special linear Lie algebra

The complex special linear Lie algebra $\mathfrak{sl}(1,\mathbb C)$ is the trace-zero subspace of the general linear Lie algebra, with bracket

\mathfrak{sl}(1,\mathbb C)=\{X\in M_{1}(\mathbb C):\operatorname{tr}X=0\},\qquad [X,Y]=XY-YX.

Dimension and structure

Trace is one nonzero linear equation, so the complex dimension is $0$ . Closure follows from $\operatorname{tr}(XY)=\operatorname{tr}(YX)$ . The trace equation forces the only entry to vanish: this is the zero Lie algebra, not a simple Lie algebra.

The family definition explains its general construction.

Scalar convention

This record retains $\mathbb C$ as its defining field. Its underlying real vector space has twice the complex dimension; real-linear maps need not be complex-linear.