On Grassmannizable Group 3-Webs

Abstract

The authors prove that the Lie group G generating a Grassmannizable group 3-web GGW is the group of parameters of the group of similarity transformations of an (r−1)-dimensional affine space \(\mathbb{A}^{r-1}\) . The transitive action of the group G on itself is an r-parameter subgroup B(r) of the group A(r 2+r) of affine transformations z I =a J I x J +b I ,I,J=1,…,r, which is the direct product of the one-dimensional group of homotheties z 1=kx 1 and r−1 one-dimensional groups of affine transformations \(z^{i}=kx^{i}+b^{i},\;i=2,\ldots ,r,\) where all r groups have the same homothety coefficient k. Conversely, the Lie group B(r) described above generates a Grassmannizable group 3-web GGW. The Lie group G is solvable but not nilpotent.