Respectful decompositions of Lie algebras

TL;DR

This paper explores the properties of respectful decompositions in Lie algebras, extending Molino's work on Riemannian foliations to non-integrable distributions, highlighting their mathematical significance.

Contribution

It introduces and analyzes the concept of respectful decompositions of Lie algebras, expanding the understanding of their structure beyond integrable cases.

Findings

Characterization of respectful decompositions

Extension of Molino's theory to non-integrable distributions

Basic properties of respectful decompositions

Abstract

One of Pierre Molino's principal mathematical achievements was his theory of Riemannian foliations. One of his last papers, published in 2001, showed that his theory could be extended to a large class of non-integrable distributions. The key example here is that of a \emph{respectful decomposition} of a Lie algebra $\mathfrak{g}$; this is vector space decomposition $\mathfrak{g}=H+V$ such that $[V,H]\subseteq H$. This paper will examine the basic properties of respectful decompositions.