Identity (mathematics): Difference between revisions

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-In [[mathematics]], the '''affine group''' or '''general affine group''' of any [[affine space]] over a [[field (mathematics)|field]] ''K'' is the [[group (mathematics)|group]] of all invertible [[affine transformation]]s from the space into itself.
+== FYI Nike Blazer High ==
-It is a [[Lie group]] if ''K'' is the real or complex field or [[quaternions]].
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-==Relation to general linear group==
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-===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'':
-:<math>\operatorname{Aff}(A) = V \rtimes \operatorname{GL}(V)</math>
-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:
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-:<math>\operatorname{Aff}(n,K) = K^n \rtimes \operatorname{GL}(n,K)</math>
-where here the natural action of GL(''n'',''K'') on ''K<sup>n</sup>'' is matrix multiplication of a vector.
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-===Stabilizer of a point===
+== Damn Ray Bans Cheap ==
-Given the affine group of an affine space ''A'', the [[Group_action#Orbits_and_stabilizers|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 <math>(A,p)</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 ''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]]
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-:<math>1 \to V \to V \rtimes \operatorname{GL}(V) \to \operatorname{GL}(V) \to 1</math>.
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-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'').
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-==Matrix representation==
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-Representing the affine group as a semidirect product of ''V'' by GL(''V''), then [[Semidirect_product#Semidirect_products_and_group_homomorphisms|by construction of the semidirect product]], the elements are pairs (''M'',&nbsp;''v''), where ''v'' is a vector in ''V'' and ''M'' is a linear transform in GL(''V''), and multiplication is given by:
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-:<math>(M,v) \cdot (N,w) = (MN, v+Mw).\,</math>
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-This can be represented as the (''n''&nbsp;+&nbsp;1)&times;(''n''&nbsp;+&nbsp;1) [[block matrix]]:
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-:<math>\left( \begin{array}{c|c} M & v\\ \hline 0 & 1 \end{array}\right) </math>
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-where ''M'' is an ''n''&times;''n'' matrix over ''K'', ''v'' an ''n''&nbsp;&times;&nbsp;1 column vector, 0 is a 1&nbsp;&times;&nbsp;''n'' row of zeros, and 1 is the 1&nbsp;&times;&nbsp;1 identity block matrix.
+ </ul>
-Formally, Aff(''V'') is naturally isomorphic to a subgroup of <math>\operatorname{GL}(V \oplus K)</math>, with ''V'' embedded as the affine plane <math>\{(v,1) | v \in V\}</math>, namely the stabilizer of this affine plane; the above matrix formulation is the (transpose of) the realization of this, with the (''n''&nbsp;&times;&nbsp;''n'' and 1&nbsp;&times;&nbsp;1) blocks corresponding to the direct sum decomposition <math>V \oplus K</math>.
-A [[matrix similarity|similar]] representation is any (''n''&nbsp;+&nbsp;1)&times;(''n''&nbsp;+&nbsp;1) matrix in which the entries in each column sum to 1.<ref>David G. Poole, "The Stochastic Group'", ''[[American Mathematical Monthly]]'', volume 102, number 9 (November, 1995), pages 798&ndash;801</ref>  The [[matrix similarity|similarity]] ''P'' for passing from the above kind to this kind is the (''n''&nbsp;+&nbsp;1)&times;(''n''&nbsp;+&nbsp;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.
-==Planar affine group==
-According to [[Rafael Artzy|Artzy]],<ref>Artzy p 94</ref> "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:
-#<math>x \mapsto a x + b y, \quad y \mapsto - b x + a y, \quad a, b \ne 0, \quad a^2 + b^2 = 1,</math>
-#<math>x \mapsto x + by, \quad y \mapsto y, \quad b \ne 0,</math>
-#<math>x \mapsto a x, \quad y \mapsto y/a, \quad a \ne 0,</math> where the coefficients ''a, b, c'', and ''d'' are real numbers."
-Case (1) corresponds to [[similarity transformation (geometry)|similarity transformation]]s which generate a [[subgroup]] of similarities.  [[Euclidean geometry]] corresponds to the subgroup of congruencies. It is characterized by [[Euclidean distance]] or [[angle]], which are [[invariant (mathematics)|invariant]] under the subgroup of rotations.
-Case (2) corresponds to [[shear mapping]]s. An important application is [[absolute time and space]] where [[Galilean transformation]]s 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 (mathematics)|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 <math>G < GL(V)</math> of the [[general linear group]],
-one can produce an affine group, sometimes denoted <math>\operatorname{Aff}(G)</math> analogously as <math>\operatorname{Aff}(G) := V \rtimes G</math>.
-More generally and abstractly, given any group ''G'' and a [[group representation|representation]] of ''G'' on a vector space ''V'',
-<math>\rho\colon G \to \operatorname{GL}(V)</math>
-one gets<ref>Since <math>\operatorname{GL}(V) < \operatorname{Aut}(V)</math>. 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'''.</ref> an associated affine group <math>V \rtimes_\rho G</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>1 \to V \to V \rtimes_\rho G \to G \to 1.</math>
-===Special affine group===
-{{main|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:.<ref>Günter Ewald (1971) ''Geometry: An Introduction'', p. 241, Belmont: Wadsworth ISBN0-534-0034-7</ref>
-:The set <math>\mathfrak{P}</math> of all projective collineations of ''P''<sup>n</sup> is a group which we may call the '''projective group''' of ''P''<sup>n</sup>. If we proceed from ''P''<sup>n</sup> to the affine space ''A''<sup>n</sup> by declaring a [[hyperplane]] &omega; to be a [[hyperplane at infinity]], we obtain the '''affine group''' <math>\mathfrak{A}</math> of ''A''<sup>n</sup> as the [[subgroup]] of <math>\mathfrak{P}</math> consisting of all elements of <math>\mathfrak{P}</math> that leave &omega; fixed.
-::<math>\mathfrak{A} \subset \mathfrak{P}</math>
-===Poincaré group===
-{{main|Poincaré group}}
-The [[Poincaré group]] is the affine group of the [[Lorentz group]] <math>O(1,3)</math>: <math>\mathbf{R}^{1,3}\rtimes \operatorname{O}(1,3)</math>
-This example is very important in [[Theory of relativity|relativity]].
-==See also==
-* [[Holomorph (mathematics)|Holomorph]]
-* [http://modis.ispras.ru/wikipedia/pic/Affine_geometry.html Related articles on a diagram]
-==References==
-{{reflist}}
-* [[Rafael Artzy]] (1965) ''Linear Geometry'', Chapter 2-6 Subgroups of the Plane Affine Group over the Real Field, [[Addison-Wesley]].
-* [[Roger Lyndon]] (1985) ''Groups and Geometry'', Section VI.1, [[Cambridge University Press]], ISBN 0-521-31694-4.
-[[Category:Affine geometry]]
-[[Category:Group theory]]
-[[Category:Lie groups]]

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