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Difference between revisions of "Affine unimodular group"

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''equi-affine group''
 
''equi-affine group''
  
The subgroup of the general [[Affine group|affine group]] consisting of the affine transformations of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011150/a0111501.png" />-dimensional affine space
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The subgroup of the general [[Affine group|affine group]] consisting of the affine transformations of the $n$-dimensional affine space
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011150/a0111502.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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x \mapsto \tilde{x} = A x + \alpha
 
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$$
that satisfy the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011150/a0111503.png" />. If the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011150/a0111504.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011150/a0111505.png" /> are interpreted as rectangular coordinates of points in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011150/a0111506.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011150/a0111507.png" />, then the transformation (*) will preserve the volumes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011150/a0111508.png" />-dimensional domains of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011150/a0111509.png" />. This makes it possible to introduce the concept of volume in an equi-affine space, which is a space with a fundamental affine unimodular group. If, in formulas (*), one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011150/a01115010.png" />, then one obtains a centro-affine unimodular group of transformations isomorphic to the group of all matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011150/a01115011.png" /> with determinant equal to one. Such a group of matrices is called the unimodular group or special linear group of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011150/a01115012.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011150/a01115014.png" />.
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that satisfy the condition $\det A = 1$. If the vectors $x$ and $\tilde{x}$ are interpreted as rectangular coordinates of points in the $n$-dimensional Euclidean space $E^n$, then the transformation (*) will preserve the volumes of $n$-dimensional domains of $E^n$. This makes it possible to introduce the concept of volume in an equi-affine space, which is a space with a fundamental affine unimodular group. If, in formulas (*), one puts $\alpha=0$, then one obtains a centro-affine unimodular group of transformations isomorphic to the group of all matrices of order $n$ with determinant equal to one. Such a group of matrices is called the unimodular group or special linear group of order $n$ and is denoted by $\mathrm{SL}(n)$.

Revision as of 18:11, 12 October 2014

equi-affine group

The subgroup of the general affine group consisting of the affine transformations of the $n$-dimensional affine space $$ x \mapsto \tilde{x} = A x + \alpha $$ that satisfy the condition $\det A = 1$. If the vectors $x$ and $\tilde{x}$ are interpreted as rectangular coordinates of points in the $n$-dimensional Euclidean space $E^n$, then the transformation (*) will preserve the volumes of $n$-dimensional domains of $E^n$. This makes it possible to introduce the concept of volume in an equi-affine space, which is a space with a fundamental affine unimodular group. If, in formulas (*), one puts $\alpha=0$, then one obtains a centro-affine unimodular group of transformations isomorphic to the group of all matrices of order $n$ with determinant equal to one. Such a group of matrices is called the unimodular group or special linear group of order $n$ and is denoted by $\mathrm{SL}(n)$.

How to Cite This Entry:
Affine unimodular group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_unimodular_group&oldid=33576
This article was adapted from an original article by A.P. Shirokov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article