Nilpotent orbits of classical Lie algebras 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.