Foliations Formed by Generic Coadjoint Orbits of Lie Groups Corresponding to a Class Seven-Dimensional Solvable Lie Algebras

TL;DR

This paper classifies the geometric and topological structures of generic coadjoint orbits in certain 7-dimensional solvable Lie groups, revealing they form measurable foliations and providing a detailed topological classification.

Contribution

It offers a comprehensive geometric and topological analysis of coadjoint orbit foliations for a specific class of 7-dimensional solvable Lie groups with a given nilradical.

Findings

Maximal-dimensional coadjoint orbits are geometrically described.

Generic orbits form measurable foliations in the sense of Connes.

A topological classification of these foliations is provided.

Abstract

We consider all connected and simply connected 7-dimensional Lie groups whose Lie algebras have nilradical $\g_{5,2} = \s \{X_1, X_2, X_3, X_4, X_5 \colon [X_1, X_2] = X_4, [X_1, X_3] = X_5\}$ of Dixmier. First, we give a geometric description of the maximal-dimensional orbits in the coadjoint representation of all considered Lie groups. Next, we prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. Finally, the topological classification of all these foliations is also provided.