Affine unimodular group

equi-affine group

The subgroup of the general affine group consisting of the affine transformations of the -dimensional affine space

(*)

that satisfy the condition . If the vectors and are interpreted as rectangular coordinates of points in the -dimensional Euclidean space , then the transformation (*) will preserve the volumes of -dimensional domains of . 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 , then one obtains a centro-affine unimodular group of transformations isomorphic to the group of all matrices of order with determinant equal to one. Such a group of matrices is called the unimodular group or special linear group of order and is denoted by .