gl(3,C) — complex general linear Lie algebra

The complex general linear Lie algebra $\mathfrak{gl}(3,\mathbb C)$ is the complex vector space of all $3\times3$ matrices over $\mathbb C$ , with commutator bracket.

\mathfrak{gl}(3,\mathbb C)=M_{3}(\mathbb C),\qquad [X,Y]=XY-YX.

Dimension and structure

The matrix units $E_{ij}$ form a basis, so its complex dimension is $9$ . The commutator of an associative matrix algebra satisfies the Jacobi identity by expansion.

Its center is the scalar matrices; the trace-zero subspace is an ideal because $\operatorname{tr}[X,Y]=0$ .

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.