Notes on a Lie algebraic relation

TL;DR

This paper explores the structure of restricted root spaces in real semi-simple Lie algebras and connects these findings to the behavior of smooth functions on G-homogeneous spaces, revealing new algebraic and geometric insights.

Contribution

It provides explicit orthogonal decompositions of multidimensional restricted root spaces and links these to the radiality of functions on homogeneous spaces, a novel connection.

Findings

Explicit orthogonal decompositions of restricted root spaces.

A demonstrated link between algebraic structure and function radiality.

Insights into the geometric properties of G-homogeneous spaces.

Abstract

We exhibit explicit orthogonal decompositions of every multidimensional restricted root space of a real semi-simple Lie algebra. We then show a link between this result and a radiality property of smooth functions on G-homogeneous spaces when G is a real semi-simple Lie group.