Aff(n,C) - Knowlpedia

For every integer $n\ge1$ , $\mathrm{Aff}(n,\mathbb C)$ is the Lie group of invertible affine maps $x\mapsto Ax+b$ of $\mathbb C^{n}$ , with $A\in\mathrm{GL}(n,\mathbb C)$ and $b\in\mathbb C^{n}$ . In coordinates,

(A,b)(A^{\prime},b^{\prime})=(AA^{\prime},Ab^{\prime}+b).

Dimensions and structure

This group has real dimension $2(n^2+n)$ , complex dimension $n^2+n$ . The translations form a normal additive subgroup. Projection to the linear part splits by $A\mapsto(A,0)$ , giving $\mathbb C^{n}\rtimes\mathrm{GL}(n,\mathbb C)$ . In dimension one this is the ax+b group.

Its complex Lie group structure uses holomorphic multiplication and inversion. Forgetting the complex structure gives the underlying real Lie group; real and holomorphic homomorphisms belong to different categories.

Direct verification

Composing the maps x ↦ Ax+b and x ↦ A′x+b′ gives x ↦ AA′x+Ab′+b. The inverse is x ↦ A⁻¹x−A⁻¹b. The underlying manifold is the product of the general linear group and the translation vector space.