Coadjoint orbits of Lie algebras and Cartan class

TL;DR

This paper explores the structure of coadjoint orbits in Lie algebras using Cartan class, identifying conditions for uniform orbit dimensions and classifying certain Lie algebras based on orbit properties.

Contribution

It introduces a novel approach linking coadjoint orbits to Cartan class and characterizes Lie algebras with specific orbit dimension patterns.

Findings

Characterization of tangent spaces to coadjoint orbits

Classification of Lie algebras with all nontrivial orbits of same dimension

Identification of Lie algebras with maximal orbit dimension in even and odd cases

Abstract

We study the coadjoint orbits of a Lie algebra in terms of Cartan class. In fact, the tangent space to a coadjoint orbit $\mathcal{O}(\alpha)$ at the point $\alpha$ corresponds to the characteristic space associated to the left invariant form;$\alpha$ and its dimension is the even part of the Cartan class of $\alpha$. We apply this remark to determine Lie algebras such that all the nontrivial orbits (nonreduced to a point) have the same dimension, in particular when this dimension is 2 or 4. We determine also the Lie algebras of dimension $2n$ or $2n+1$ having an orbit of dimension $2n$.