Affine transformation Lie algebra aff(n,C)

For an integer $n\geq1$ , the complex affine transformation Lie algebra $\mathfrak{aff}(n,\mathbb C)$ is $M_{n}(\mathbb C)\oplus \mathbb C^{n}$ with bracket

[(A,v),(B,w)]=(AB-BA,Aw-Bv).

It describes infinitesimal affine transformations of $\mathbb C^{n}$ .

Matrix model and convention

The block-matrix embedding

(A,v)\longmapsto\begin{pmatrix}A&v\\0&0\end{pmatrix}

turns the displayed bracket into a matrix commutator, proving the Lie axioms. The translation vectors form an abelian ideal, and the quotient is the general linear Lie algebra.

This finite-dimensional algebra of affine transformations is not an affine Kac–Moody algebra.