Representation of real solvable Lie algebras having 2-dimensional derived ideal and geometry of coadjoint orbits of corresponding Lie groups

Abstract

Let Lie ([Formula: see text]) be the class of all [Formula: see text]-dimensional real solvable Lie algebras having [Formula: see text]-dimensional derived ideals. In 2020, Le et al. gave a classification of all non-2-step nilpotent Lie algebras of Lie ([Formula: see text], 2). We propose in this paper to study representations of these Lie algebras as well as their corresponding connected and simply connected Lie groups. That is, for each algebra, we give an upper bound of the minimal degree of a faithful representation. Then, we give a geometrical description of coadjoint orbits of corresponding groups. Moreover, we show that the characteristic property of the family of maximal dimensional coadjoint orbits of an MD-group studied by Shum et al. is still true for the Lie groups considered here. Namely, we prove that, for each considered group, the family of the maximal dimensional coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.