View source for Affine unimodular 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

<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>

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" />.