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} (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:

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

where here the natural action of $GL(n, K)$ on $K n$ 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+Mw)\,.

This can be represented as the $(n + 1) \times (n + 1)$ block matrix:

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

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

Formally, $Aff(V)$ is naturally isomorphic to a subgroup of $GL(V \oplus K)$ , 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 $n \times n$ and $1 \times 1$ ) blocks corresponding to the direct sum decomposition $V \oplus K$ .

A similar representation is any $(n + 1) \times (n + 1)$ 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 $(n + 1) \times (n + 1)$ 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

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

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

Character table of $Aff(F p)$

$Aff(F p)$ has order $p (p - 1)$ . Since

{\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 $Aff(F p)$ has $p$ conjugacy classes, namely

{\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 $Aff(F p)$ has $p$ irreducible representations. By above paragraph (§ Matrix representation), there exist $p - 1$ one-dimensional representations, decided by the homomorphism

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

for $k = 1, 2,\dots p - 1$ , where

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

and $i 2 = -1$ , $a = g j$ , $g$ is a generator of the group $F * p$ . Then compare with the order of $F p$ , we have

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

hence $χ p = p - 1$ is the dimension of the last irreducible representation. Finally using the orthogonality of irreducible representations, we can complete the character table of $Aff(F p)$ :

{\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:

{\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 $Aff(G) := V ⋊ G$ .

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

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

one gets[note 1] an associated affine group $V ⋊ ρ G$ : 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

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

{\mathfrak {P}}

of all projective collineations of

P n

is a group which we may call the projective group of

P n

. If we proceed from

P n

to the affine space

A n

by declaring a hyperplane

ω

to be a hyperplane at infinity, we obtain the affine group

{\mathfrak {A}}

of

A n

as the subgroup of

{\mathfrak {P}}

consisting of all elements of

{\mathfrak {P}}

that leave

ω

fixed.

{\mathfrak {A}}\subset {\mathfrak {P}}

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.

Notes

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

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

Lyndon, Roger (1985). "Section VI.1". Groups and Geometry. Cambridge University Press. ISBN 0-521-31694-4.

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[3] 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 $R$ .

[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.

[4] Ewald, Günter (1971). Geometry: An Introduction. Belmont: Wadsworth. p. 241. ISBN 9780534000349.