sl(1,R) — real special linear Lie algebra

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

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

Dimension and structure

Trace is one nonzero linear equation, so the real 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 $\mathbb R$ as its defining field. Its complexification is a different catalogue object; a matrix presentation over complex numbers does not change this field automatically.