Representation of Real Solvable Lie Algebras Having 2-dimensional Derived Ideal and Geometry of Coadjoint Orbits of Corresponding Lie Groups

TL;DR

This paper classifies certain 2-dimensional derived ideal solvable Lie algebras, studies their representations, describes their coadjoint orbits geometrically, and analyzes the foliation structure of maximal orbits.

Contribution

It provides a detailed study of representations and coadjoint orbit geometry for a specific class of solvable Lie algebras with 2-dimensional derived ideals, extending previous classifications.

Findings

Upper bounds for minimal faithful representation degrees.

Geometric description of coadjoint orbits.

Maximal coadjoint orbits form measurable foliations.

Abstract

Let {\Lnk} be the class of all $n$-dimensional real solvable Lie algebras having $k$-dimensional derived ideals. In 2020 the authors et al. gave a classification of all non 2-step nilpotent Lie algebras of {\Li}. 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 a MD-group studied by K. P. Shum and the second author 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.