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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 $n$-dimensional affine space
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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
 
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 (cf. [[Equi-affine geometry]]). 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 or [[special linear group]] of order $n$ and is denoted by $\mathrm{SL}(n)$.
 
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 (cf. [[Equi-affine geometry]]). 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 or [[special linear group]] of order $n$ and is denoted by $\mathrm{SL}(n)$.
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[[Category:Group theory and generalizations]]
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[[Category:Geometry]]

Latest revision as of 22:38, 2 November 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 (cf. Equi-affine geometry). 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 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=33577
This article was adapted from an original article by A.P. Shirokov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article