gl(1,C) — complex general linear Lie algebra

The complex general linear Lie algebra $\mathfrak{gl}(1,\mathbb C)$ is the complex vector space of all $1\times1$ matrices over $\mathbb C$ , with commutator bracket.

\mathfrak{gl}(1,\mathbb C)=M_{1}(\mathbb C),\qquad [X,Y]=XY-YX.

Dimension and structure

The matrix units $E_{ij}$ form a basis, so its complex dimension is $1$ . The commutator of an associative matrix algebra satisfies the Jacobi identity by expansion.

In size one every pair of matrices commutes, so this is the one-dimensional abelian 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.