Affine group

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 is the real or complex field, then the affine group is a Lie group.

Relation to general linear group

Construction from general linear group

Concretely, given a vector space Template:Mvar, it has an underlying affine space Template:Mvar obtained by "forgetting" the origin, with Template:Mvar acting by translations, and the affine group of Template:Mvar can be described concretely as the semidirect product of Template:Mvar by Template:Math, the general linear group of Template:Mvar:

<math>\operatorname{Aff}(V) = V \rtimes \operatorname{GL}(V)</math>

The action of Template:Math on Template:Mvar is the natural one (linear transformations are automorphisms), so this defines a semidirect product.

In terms of matrices, one writes:

<math>\operatorname{Aff}(n,K) = K^n \rtimes \operatorname{GL}(n,K)</math>

where here the natural action of Template:Math on Template:Mvar is matrix multiplication of a vector.

Stabilizer of a point

Given the affine group of an affine space Template:Mvar, the stabilizer of a point Template:Mvar is isomorphic to the general linear group of the same dimension (so the stabilizer of a point in Template:Math is isomorphic to Template:Math); formally, it is the general linear group of the vector space Template:Math: 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 Template:Mvar to Template:Mvar (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

<math>1 \to V \to V \rtimes \operatorname{GL}(V) \to \operatorname{GL}(V) \to 1\,.</math>

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 Template:Math.

Matrix representation

Representing the affine group as a semidirect product of Template:Mvar by Template:Math, then by construction of the semidirect product, the elements are pairs Template:Math, where Template:Mvar is a vector in Template:Mvar and Template:Mvar is a linear transform in Template:Math, and multiplication is given by

<math>(v, M) \cdot (w, N) = (v+Mw, MN)\,.</math>

This can be represented as the Template:Math block matrix

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

where Template:Mvar is an Template:Math matrix over Template:Mvar, Template:Mvar an Template:Math column vector, 0 is a Template:Math row of zeros, and 1 is the Template:Nowrap identity block matrix.

Formally, Template:Math is naturally isomorphic to a subgroup of Template:Math, with Template:Mvar embedded as the affine plane Template:Math, namely the stabilizer of this affine plane; the above matrix formulation is the (transpose of the) realization of this, with the Template:Math and Template:Math blocks corresponding to the direct sum decomposition Template:Math.

A similar representation is any Template:Math matrix in which the entries in each column sum to 1.<ref>Template:Cite journal</ref> The similarity Template:Mvar for passing from the above kind to this kind is the Template:Math 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 Template:Math, that is, the upper triangular Template:Nowrap 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), Template:Mvar and Template:Mvar, such that Template:Math, where

<math> 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)\,,</math>

so that

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

Character table of Template:Math

Template:Math has order Template:Math. Since

<math>\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}\,,</math>

we know Template:Math has Template:Mvar conjugacy classes, namely

<math>\begin{align}

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{align}</math>

Then we know that Template:Math has Template:Mvar irreducible representations. By above paragraph (Template:Section link), there exist Template:Math one-dimensional representations, decided by the homomorphism

<math>\rho_k:\operatorname{Aff}(\mathbf{F}_p)\to\Complex^*</math>

for Template:Math, where

<math>\rho_k\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix}=\exp\left(\frac{2i kj\pi}{p-1}\right)</math>

and Template:Math, Template:Math, Template:Mvar is a generator of the group Template:Math. Then compare with the order of Template:Math, we have

<math>p(p-1)=p-1+\chi_p^2\,,</math>

hence Template:Math is the dimension of the last irreducible representation. Finally using the orthogonality of irreducible representations, we can complete the character table of Template:Math:

<math>\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}</math>

Planar affine group over the reals

The elements of <math>\operatorname{Aff}(2,\mathbb R)</math> can take a simple form on a well-chosen affine coordinate system. More precisely, given an affine transformation of an affine plane over the reals, an affine coordinate system exists on which it has one of the following forms, where Template:Mvar, Template:Mvar, and Template:Mvar are real numbers (the given conditions insure that transformations are invertible, but not for making the classes distinct; for example, the identity belongs to all the classes).

<math>\begin{align}

\text{1.}&& (x, y) &\mapsto (x +a,y+b),\\[3pt] \text{2.}&& (x, y) &\mapsto (ax,by), &\qquad \text{where } ab\ne 0,\\[3pt] \text{3.}&& (x, y) &\mapsto (ax,y+b), &\qquad \text{where } a\ne 0,\\[3pt] \text{4.}&& (x, y) &\mapsto (ax+y,ay), &\qquad \text{where } a\ne 0,\\[3pt] \text{5.}&& (x, y) &\mapsto (x+y,y+a)\\[3pt] \text{6.}&& (x, y) &\mapsto (a(x\cos t + y\sin t), a(-x\sin t+y\cos t)), &\qquad \text{where } a\ne 0. \end{align}</math>

Case 1 corresponds to translations.

Case 2 corresponds to scalings that may differ in two different directions. When working with a Euclidean plane these directions need not be perpendicular, since the coordinate axes need not be perpendicular.

Case 3 corresponds to a scaling in one direction and a translation in another one.

Case 4 corresponds to a shear mapping combined with a dilation.

Case 5 corresponds to a shear mapping combined with a dilation.

Case 6 corresponds to similarities when the coordinate axes are perpendicular.

The affine transformations without any fixed point belong to cases 1, 3, and 5. The transformations that do not preserve the orientation of the plane belong to cases 2 (with Template:Math) or 3 (with Template:Math).

The proof may be done by first remarking that if an affine transformation has no fixed point, then the matrix of the associated linear map has an eigenvalue equal to one, and then using the Jordan normal form theorem for real matrices.

Other affine groups and subgroups

General case

Given any subgroup Template:Math of the general linear group, one can produce an affine group, sometimes denoted Template:Math, analogously as Template:Math.

More generally and abstractly, given any group Template:Mvar and a representation <math>\rho : G \to \operatorname{GL}(V)</math> of Template:Mvar on a vector space Template:Mvar, one gets<ref group="note">Since Template:Math. Note that this containment is in general proper, since by "automorphisms" one means group automorphisms, i.e., they preserve the group structure on Template:Mvar (the addition and origin), but not necessarily scalar multiplication, and these groups differ if working over Template:Math.</ref> an associated affine group Template:Math: 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 <math display="block">1 \to V \to V \rtimes_\rho G \to G \to 1.</math>

Special affine group

The subset of all invertible affine transformations that preserve a fixed volume form up to sign is called the special affine group. (The transformations themselves are sometimes called equiaffinities.) This group is the affine analogue of the special linear group. In terms of the semi-direct product, the special affine group consists of all pairs Template:Math with <math>|\det(M)| = 1</math>, that is, the affine transformations <math display="block">x \mapsto Mx + v</math> where Template:Mvar is a linear transformation of whose determinant has absolute value 1 and Template:Mvar is any fixed translation vector.<ref>Template:Cite book</ref><ref>Template:Cite book</ref>

The subgroup of the special affine group consisting of those transformations whose linear part has determinant 1 is the group of orientation- and volume-preserving maps. Algebraically, this group is a semidirect product <math>SL(V) \ltimes V</math> of the special linear group of <math>V</math> with the translations. It is generated by the shear mappings.

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:<ref>Template:Cite book</ref>

The set <math>\mathfrak{P}</math> of all projective collineations of Template:Math is a group which we may call the projective group of Template:Math. If we proceed from Template:Math to the affine space Template:Math by declaring a hyperplane Template:Mvar to be a hyperplane at infinity, we obtain the affine group <math>\mathfrak{A}</math> of Template:Math as the subgroup of <math>\mathfrak{P}</math> consisting of all elements of <math>\mathfrak{P}</math> that leave Template:Mvar fixed.

<math>\mathfrak{A} \subset \mathfrak{P}</math>

Isometries of Euclidean space

When the affine space Template:Mvar is a Euclidean space (over the field of real numbers), the group <math>\mathcal{E}</math> of distance-preserving maps (isometries) of Template:Mvar is a subgroup of the affine group. Algebraically, this group is a semidirect product <math>O(V) \ltimes V</math> of the orthogonal group of <math>V</math> with the translations. Geometrically, it is the subgroup of the affine group generated by the orthogonal reflections.

Poincaré group

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The Poincaré group is the affine group of the Lorentz group Template:Math:

<math>\mathbf{R}^{1,3}\rtimes \operatorname{O}(1,3)</math>

This example is very important in relativity.

See also

• Affine Coxeter group – certain discrete subgroups of the affine group on a Euclidean space that preserve a lattice
• Holomorph

References

• Template:Cite book