Affine group

In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.

It is a Lie group if K is the real or complex field or quaternions.

Contents

Relation to general linear group

Construction from general linear 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:

\operatorname{Aff}(A) = V \rtimes \operatorname{GL}(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:

\operatorname{Aff}(n,K) = K^n \rtimes \operatorname{GL}(n,K)

where here the natural action of GL(n,K) on Kn is matrix multiplication of a vector.

Stabilizer of a point

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

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 GL(V).

Matrix representation

Representing the affine group as a semidirect product of V by GL(V), then by construction of the semidirect product, the elements are pairs (M, v), where v is a vector in V and M is a linear transform in GL(V), and multiplication is given by:

(M,v) \cdot (N,w) = (MN, v%2BMw).\,

This can be represented as the (n + 1)×(n + 1) block matrix:

\left( \begin{array}{c|c} M & v\\ \hline 0 & 1 \end{array}\right)

where M is an n×n matrix over K, v an n × 1 column vector, 0 is a 1 × n row of zeros, and 1 is the 1 × 1 identity block matrix.

Formally, Aff(V) is naturally isomorphic to a subgroup of \operatorname{GL}(V \oplus K), with V embedded as the affine plane \{(v,1) | v \in V\}, namely the stabilizer of this affine plane; the above matrix formulation is the (transpose of) the realization of this, with the (n × n and 1 × 1) blocks corresponding to the direct sum decomposition V \oplus K.

A similar representation is any (n + 1)×(n + 1) matrix in which the entries in each column sum to 1.[1] The similarity P for passing from the above kind to this kind is the (n + 1)×(n + 1) identity matrix with the bottom row replaced by a row of all ones.

Each of these two classes of matrices is closed under matrix multiplication.

Other affine groups

General case

Given any subgroup G < GL(V) of the general linear group, one can produce an affine group, sometimes denoted \operatorname{Aff}(G) analogously as \operatorname{Aff}(G)�:= V \rtimes G.

More generally and abstractly, given any group G and a representation of G on a vector space V, \rho\colon G \to \operatorname{GL}(V) one gets[2] an associated affine group V \rtimes_\rho G: one can say that the affine group obtained is “a group extension by a vector representation”, and as above, one has the short exact sequence:

1 \to V \to V \rtimes_\rho G \to G \to 1.

Special affine group

Main article: Special affine group

The subset of all invertible affine transformations preserving a fixed volume form, or in terms of the semi-direct product, the set of all elements (M,v) with M of determinant 1, is a subgroup known as the special affine group.

Poincaré group

Main article: Poincaré group

The Poincaré group is the affine group of the Lorentz group O(1,3): \mathbf{R}^{1,3}\rtimes \operatorname{O}(1,3)

This example is very important in relativity.

References

  1. ^ David G. Poole, "The Stochastic Group'", American Mathematical Monthly, volume 102, number 9 (November, 1995), pages 798–801
  2. ^ Since \operatorname{GL}(V) < \operatorname{Aut}(V). Note that this containment is in general proper, since by “automorphisms” one means group automorphisms, i.e., they preserve the group structure on V (the addition and origin), but not necessarily scalar multiplication, and these groups differ if working over R.