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Concept
Affine group
Summary
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 the real or complex field, then the affine group is a Lie group.
Concretely, given a vector space V, it has an underlying affine space A obtained by "forgetting" the origin, with V acting by translations, and the affine group of A can be described concretely as the semidirect product of V by GL(V), the general linear group of V:
The action of GL(V) on V is the natural one (linear transformations are automorphisms), so this defines a semidirect product.
In terms of matrices, one writes:
where here the natural action of GL(n, K) on Kn is matrix multiplication of a vector.
Given the affine group of an affine space A, the stabilizer of a point p is isomorphic to the general linear group of the same dimension (so the stabilizer of a point in Aff(2, R) is isomorphic to GL(2, R)); formally, it is the general linear group of the vector space (A, p): 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 p to q (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
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 GL(V).
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