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.

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"::$operatorname\{Aff\}(A)\; =\; V\; times\; operatorname\{GL\}(V)$The action of GL("V") on "V" is the natural one (linear transforms are automorphisms), so this defines a semidirect product.

In terms of matrices, one writes::$operatorname\{Aff\}(n,K)\; =\; K^n\; times\; operatorname\{GL\}(n,K)$where here the natural action of GL("n","K") on "Kⁿ" is matrix multiplication of a vector.

tabilizer 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:$1\; o\; V\; o\; V\; times\; operatorname\{GL\}(V)\; o\; operatorname\{GL\}(V)\; o\; 1$.

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:

:$(M,v)\; cdot\; (N,w)\; =\; (MN,\; v+Mw).,$

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

:

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. [David G. Poole, "The Stochastic Group'", "American Mathematical Monthly", volume 102, number 9 (November, 1995), pages 798–801] 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 of the general linear group,one can produce an affine group, sometimes denoted $operatorname\{Aff\}(G)$ analogously as $operatorname\{Aff\}(G)\; :=\; V\; times\; G$.

More generally and abstractly, given any group "G" and a representation of "G" on a vector space "V",$hocolon\; G\; o\; operatorname\{GL\}(V)$one gets [Since . 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.] an associated affine group $V\; times\_\; ho\; G$: 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::$1\; o\; V\; o\; V\; times\_\; ho\; G\; o\; G\; o\; 1.$

pecial 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

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

This example is very important in relativity.

References

* R.C. Lyndon, "Groups and Geometry", Cambridge University Press, 1985, ISBN 0-521-31694-4. Section VI.1.