gl(1,R) — real general linear Lie algebra

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

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

Dimension and structure

The matrix units $E_{ij}$ form a basis, so its real 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 $\mathbb R$ as its defining field. Its complexification is a different catalogue object; a matrix presentation over complex numbers does not change this field automatically.