Foliations formed by generic coadjoint orbits of a class of 7-dimensional real solvable Lie groups

TL;DR

This paper studies the geometric structure of generic coadjoint orbits of certain 7-dimensional solvable Lie groups, showing they form measurable foliations and providing their topological classification.

Contribution

It describes the geometry of coadjoint orbits for specific 7D solvable Lie groups and classifies the resulting foliations topologically.

Findings

Generic coadjoint orbits form measurable foliations

Topological classification of these foliations provided

Description of orbit geometry for specific Lie groups

Abstract

In this paper, we consider exponential, connected and simply connected Lie groups which are corresponding to Lie algebras of dimension 7 such that the nilradical of them is 5-dimensional nilpotent Lie algebra $\mathfrak{g}_{5,2}$ in Table 1. In particular, we give a description of the geometry of the generic orbits in the coadjoint representation of some considered Lie groups. We prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.