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

For an integer $n\geq1$ , the complex general linear Lie algebra $\mathfrak{gl}(n,\mathbb C)$ is the complex vector space of all $n\times n$ matrices over $\mathbb C$ , with commutator bracket.

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

Dimension and structure

The matrix units $E_{ij}$ form a basis, so its complex dimension is $n^2$ . 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.