Nilpotent orbits of classical Lie algebras stable under negation

TL;DR

This paper investigates nilpotent coadjoint orbits of classical Lie algebras, showing Gibbs states do not exist on orbits stable under negation, and provides a classification of such orbits.

Contribution

It proves the non-existence of Gibbs states on certain nilpotent orbits and classifies all classical Lie algebra orbits stable under negation.

Findings

Gibbs states do not exist on nilpotent orbits stable under negation.

Classification of all classical Lie algebra nilpotent orbits stable under negation.

Abstract

Gibbs states are probability distributions defined on Hamiltonian G-manifolds that are naturally parametrized by elements of the Lie algebra g. In this paper, we focus on a specific case of the simplest Hamiltonian G-manifolds, the coadjoint orbits of Lie algebras. We look at the nilpotent coadjoint orbits of the classical Lie algebras, or equivalently the nilpotent adjoint orbits. We show that Gibbs states do not exist on nilpotent orbits that are stable under multiplication by -1, and proceed to classify those for all classical Lie algebras.