On Linear Homogeneous Hypersurfaces in ${{\mathbb{R}}^{4}}$

Abstract

The problem of describing affinely homogeneous hypersurfaces in the space ${{\mathbb{R}}^{4}}$ that possess exactly three-dimensional algebras of affine symmetries is discussed. For three types of solvable three-dimensional Lie algebras, their linearly homogeneous three-dimensional orbits in this space that differ from the second-order and cylindrical surfaces in ${{\mathbb{R}}^{4}}$ (which are of no interest in the problem involved) are studied. The fact that there are two nontrivial commutation relations in each of the algebras involved makes the situation with their orbits in ${{\mathbb{R}}^{4}}$ differ significantly from the case of a three-dimensional Abelian algebra with a big family of affinely distinct (linearly homogeneous) orbits in the same space. One of the studied types of Lie algebras is proved to not allow for nontrivial four-dimensional linear representations at all, with a great number of three-dimensional orbits of representations of the two other types having rich symmetry algebras. At the same time, for one of the three types of Lie algebras, a new family of linearly homogeneous orbits that have precisely three-dimensional algebras of affine symmetries is obtained. Proofs of the propositions significantly rely on symbolic calculations associated with $(4 \times 4)$-matrices from the matrix Lie algebras involved.