Affine unimodular group

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 $\mathrm{SL}(n)$.