Affine distributions on a four-dimensional extension of the semi-Euclidean group (Q2831414)

Starting from the work by \textit{V. I. Elkin} [J. Math. Sci., New York 88, No. 5, 675--722 (1998; Zbl 0953.93020)], affine distributions have been considered by several authors in the last decades. In this paper, the authors deal with these objects and classify the invariant affine distributions (up to group automorphism) on a four-dimensional central extension of the semi-Euclidean group. Indeed, they consider the algebra \(\mathfrak g_{4,8}^{-1}\), which they denote by \(\mathfrak e_{1,1}^{\infty}\), and its associated simply connected Lie group \(\mathbf E_{1,1}^\infty\) and study the equivalence of left-invariant affine distributions on this last one. They classify the vector subspaces of \(\mathfrak e_{1,1}^\infty\) and also provide a characterization in terms of Lie algebras automorphisms of the equivalence relation between two distributions by a group automorphism. Finally, they classify the invariant affine distributions on \(\mathbf E_{1,1}^\infty\) and present two extensive examples interpreting that classification in the context of control theory and sub-Riemannian geometry.