Affine group explained

In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real numbers), the affine group consists of those functions from the space to itself such that the image of every line is a line.

Over any field, the affine group may be viewed as a matrix group in a natural way. If the associated field of scalars is the real or complex field, then the affine group is a Lie group.

Relation to general linear group

Construction from general linear group

Concretely, given a vector space, it has an underlying affine space obtained by "forgetting" the origin, with acting by translations, and the affine group of can be described concretely as the semidirect product of by, the general linear group of :

\operatorname{Aff}(V) = V \rtimes \operatorname{GL}(V) The action of on is the natural one (linear transformations are automorphisms), so this defines a semidirect product.

In terms of matrices, one writes:

\operatorname{Aff}(n,K) = K ^{n} \rtimes \operatorname{GL}(n,K) where here the natural action of on is matrix multiplication of a vector.

Stabilizer of a point

Given the affine group of an affine space, the stabilizer of a point is isomorphic to the general linear group of the same dimension (so the stabilizer of a point in is isomorphic to); formally, it is the general linear group of the vector space : recall that if one fixes a point, an affine space becomes a vector space.

All these subgroups are conjugate, where conjugation is given by translation from to (which is uniquely defined), however, no particular subgroup is a natural choice, since no point is special – this corresponds to the multiple choices of transverse subgroup, or splitting of the short exact sequence

1 \to V \to V \rtimes \operatorname{GL}(V) \to \operatorname{GL}(V) \to 1 .

In the case that the affine group was constructed by starting with a vector space, the subgroup that stabilizes the origin (of the vector space) is the original .

Matrix representation

Representing the affine group as a semidirect product of by, then by construction of the semidirect product, the elements are pairs, where is a vector in and is a linear transform in, and multiplication is given by

(v, M) ⋅ (w, N) = (v + Mw, MN).

This can be represented as the block matrix

\left( \begin{array}{c|c} M & v\ \hline 0 & 1 \end{array}\right) where is an matrix over, an column vector, 0 is a row of zeros, and 1 is the identity block matrix.

Formally, is naturally isomorphic to a subgroup of, with embedded as the affine plane