# 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:

${\displaystyle \operatorname {Aff} (V)=V\rtimes \operatorname {GL} (V)}$

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:

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

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

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

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:

${\displaystyle (M,v)\cdot (N,w)=(MN,v+Mw)\,.}$

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

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

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 GL(VK), with V embedded as the affine plane {(v, 1) | vV}, 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 VK.

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.

The simplest paradigm may well be the case n = 1, that is, the upper triangular 2 × 2 matrices representing the affine group in one dimension. It is a two-parameter non-Abelian Lie group, so with merely two generators (Lie algebra elements), A and B, such that [A, B] = B, where

${\displaystyle A=\left({\begin{array}{cc}1&0\\0&0\end{array}}\right),\qquad B=\left({\begin{array}{cc}0&1\\0&0\end{array}}\right)\,,}$

so that

${\displaystyle e^{aA+bB}=\left({\begin{array}{cc}e^{a}&{\tfrac {b}{a}}(e^{a}-1)\\0&1\end{array}}\right)\,.}$

## Character table of Aff(Fp)

Aff(Fp) has order p(p − 1). Since

${\displaystyle {\begin{pmatrix}c&d\\0&1\end{pmatrix}}{\begin{pmatrix}a&b\\0&1\end{pmatrix}}{\begin{pmatrix}c&d\\0&1\end{pmatrix}}^{-1}={\begin{pmatrix}a&(1-a)d+bc\\0&1\end{pmatrix}}\,,}$

we know Aff(Fp) has p conjugacy classes, namely

{\displaystyle {\begin{aligned}C_{id}&=\left\{{\begin{pmatrix}1&0\\0&1\end{pmatrix}}\right\}\,,\\[6pt]C_{1}&=\left\{{\begin{pmatrix}1&b\\0&1\end{pmatrix}}{\Bigg |}b\in \mathbf {F} _{p}^{*}\right\}\,,\\[6pt]{\Bigg \{}C_{a}&=\left\{{\begin{pmatrix}a&b\\0&1\end{pmatrix}}{\Bigg |}b\in \mathbf {F} _{p}\right\}{\Bigg |}a\in \mathbf {F} _{p}\setminus \{0,1\}{\Bigg \}}\,.\end{aligned}}}

Then we know that Aff(Fp) has p irreducible representations. By above paragraph (§ Matrix representation), there exist p − 1 one-dimensional representations, decided by the homomorphism

${\displaystyle \rho _{k}:\operatorname {Aff} (\mathbf {F} _{p})\to \mathbb {C} ^{*}}$

for k = 1, 2,… p − 1, where

${\displaystyle \rho _{k}{\begin{pmatrix}a&b\\0&1\end{pmatrix}}=\exp \left({\frac {2ikj\pi }{p-1}}\right)}$

and i2 = −1, a = gj, g is a generator of the group F
p
. Then compare with the order of Fp, we have

${\displaystyle p(p-1)=p-1+\chi _{p}^{2}\,,}$

hence χp = p − 1 is the dimension of the last irreducible representation. Finally using the orthogonality of irreducible representations, we can complete the character table of Aff(Fp):

${\displaystyle {\begin{array}{c|cccccc}&{\color {Blue}C_{id}}&{\color {Blue}C_{1}}&{\color {Blue}C_{g}}&{\color {Blue}C_{g^{2}}}&{\color {Gray}\dots }&{\color {Blue}C_{g^{p-2}}}\\\hline {\color {Blue}\chi _{1}}&{\color {Gray}1}&{\color {Gray}1}&{\color {Blue}e^{\frac {2\pi i}{p-1}}}&{\color {Blue}e^{\frac {4\pi i}{p-1}}}&{\color {Gray}\dots }&{\color {Blue}e^{\frac {2\pi (p-2)i}{p-1}}}\\{\color {Blue}\chi _{2}}&{\color {Gray}1}&{\color {Gray}1}&{\color {Blue}e^{\frac {4\pi i}{p-1}}}&{\color {Blue}e^{\frac {8\pi i}{p-1}}}&{\color {Gray}\dots }&{\color {Blue}e^{\frac {4\pi (p-2)i}{p-1}}}\\{\color {Blue}\chi _{3}}&{\color {Gray}1}&{\color {Gray}1}&{\color {Blue}e^{\frac {6\pi i}{p-1}}}&{\color {Blue}e^{\frac {12\pi i}{p-1}}}&{\color {Gray}\dots }&{\color {Blue}e^{\frac {6\pi (p-2)i}{p-1}}}\\{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }\\{\color {Blue}\chi _{p-1}}&{\color {Gray}1}&{\color {Gray}1}&{\color {Gray}1}&{\color {Gray}1}&{\color {Gray}\dots }&{\color {Gray}1}\\{\color {Blue}\chi _{p}}&{\color {Gray}p-1}&{\color {Gray}-1}&{\color {Gray}0}&{\color {Gray}0}&{\color {Gray}\dots }&{\color {Gray}0}\end{array}}}$

## Planar affine group

According to Rafael Artzy,[2] "The linear part of each affinity [of the real affine plane] can be brought into one of the following standard forms by a coordinate transformation followed by a dilation from the origin:

{\displaystyle {\begin{aligned}{\text{1.}}&&x&\mapsto ax+by\,,&y&\mapsto -bx+ay\,,&a,b&\neq 0\,,&a^{2}+b^{2}=1\,,\\[3pt]{\text{2.}}&&x&\mapsto x+by\,,&y&\mapsto y\,,&b&\neq 0\,,&\\[3pt]{\text{3.}}&&x&\mapsto ax\,,&y&\mapsto {\tfrac {y}{a}}\,,&a&\neq 0\,,&\end{aligned}}}

where the coefficients a, b, c, and d are real numbers."

Case 1 corresponds to similarity transformations which generate a subgroup of similarities.  Euclidean geometry corresponds to the subgroup of congruences. It is characterized by Euclidean distance or angle, which are invariant under the subgroup of rotations.

Case 2 corresponds to shear mappings. An important application is absolute time and space where Galilean transformations relate frames of reference. They generate the Galilean group.

Case 3 corresponds to squeeze mapping. These transformations generate a subgroup, of the planar affine group, called the Lorentz group of the plane. The geometry associated with this group is characterized by hyperbolic angle, which is a measure that is invariant under the subgroup of squeeze mappings.

Using the above matrix representation of the affine group on the plane, the matrix M is a 2 × 2 real matrix. Accordingly, a non-singular M must have one of three forms that correspond to the trichotomy of Artzy.

## Other affine groups

### General case

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

More generally and abstractly, given any group G and a representation of G on a vector space V,

${\displaystyle \rho :G\to \operatorname {GL} (V)}$

one gets[note 1] an associated affine group Vρ 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:

${\displaystyle 1\to V\to V\rtimes _{\rho }G\to G\to 1\,.}$

### 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.

### Projective subgroup

Presuming knowledge of projectivity and the projective group of projective geometry, the affine group can be easily specified. For example, Günter Ewald wrote:[3]

The set ${\displaystyle {\mathfrak {P}}}$  of all projective collineations of Pn is a group which we may call the projective group of Pn. If we proceed from Pn to the affine space An by declaring a hyperplane ω to be a hyperplane at infinity, we obtain the affine group ${\displaystyle {\mathfrak {A}}}$  of An as the subgroup of ${\displaystyle {\mathfrak {P}}}$  consisting of all elements of ${\displaystyle {\mathfrak {P}}}$  that leave ω fixed.
${\displaystyle {\mathfrak {A}}\subset {\mathfrak {P}}}$

### Poincaré group

The Poincaré group is the affine group of the Lorentz group O(1,3):

${\displaystyle \mathbf {R} ^{1,3}\rtimes \operatorname {O} (1,3)}$

This example is very important in relativity.

## Notes

1. ^ Since GL(V) < 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.

## References

1. ^ Poole, David G. (November 1995). "The Stochastic Group". American Mathematical Monthly. 102 (9): 798–801.
2. ^ Artzy, Rafael (1965). "Chapter 2-6: Subgroups of the Plane Affine Group over the Real Field". Linear Geometry. Addison-Wesley. p. 94.
3. ^ Ewald, Günter (1971). Geometry: An Introduction. Belmont: Wadsworth. p. 241. ISBN 9780534000349.