K-theory for the Leaf Spaces of the Orbit Foliations of the co-adjoint Action of some 5-dimensional Solvable Lie groups

TL;DR

This paper classifies the leaf space topology of certain 5-dimensional solvable Lie group orbit foliations and analyzes their K-theory and Connes' C*-algebras using advanced geometric and algebraic methods.

Contribution

It provides a classification of MD(5,3C)-foliations and characterizes their associated C*-algebras through K-theory and KK-functors, combining Kirillov's and Connes' methods.

Findings

Classification of MD(5,3C)-foliations based on co-adjoint orbits

Description of Connes' C*-algebras via KK-functors

Analysis of leaf space topology for specific Lie group actions

Abstract

In this paper, combining Kirillov's method of orbits with Connes' method in Differential Geometry, we study the so-called MD(5,3C)-foliations, i.e. the orbit foliations of the co-adjoint action of MD(5,3C)-groups. First, we classify topologically MD(5,3C)-foliations based on the classification of all MD(5,3C)-algebras in [22] and the picture of co-adjoint orbits (K-orbits) of all MD(5,3C)-groups in [23]. Finally, we study K-theory for leaf space of MD(5,3C)-foliations and describe analytically or characterize Connes' C*-algebras of the considered foliations by KK-functors.