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Nilpotent orbits of classical Lie algebras stable under negation

Guillaume Neuttiens, Jérémie Pierard de Maujouy

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.

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.

Paper Structure

This paper contains 17 sections, 29 theorems, 45 equations, 1 table.

Table of Contents

  1. Introduction
  2. Nilpotent adjoint orbits of semisimple Lie groups
  3. Standard triples
  4. Conical structure of nilpotent orbits
  5. Gibbs states
  6. Invariant products on $\sl(2, \mathbb{R})$-modules
  7. Irreducible $\sl(2, \mathbb{R})$-modules
  8. General $\sl(2, \mathbb{R})$-modules with invariant product
  9. Nilpotent orbits of the classical Lie algebras
  10. $\sl(n, \mathbb{R})$
  11. $\mathfrak{sp}(p,q)$
  12. $\mathfrak{sp}(2n, \mathbb{R})$
  13. $\operatorname{\mathfrak{so}}^*(2n)$
  14. $\mathfrak{su}(p,q)$
  15. $\sl(n, \mathbb{H})$
  16. ...and 2 more sections

Key Result

Theorem 2.2

Let $\mathfrak g$ be a real semisimple Lie algebra. For any nonzero nilpotent element $x\in \mathfrak g$, there exist $y,h\in \mathfrak g$ such that $(x, y, h)$ is a standard triple for $\mathfrak g$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (58)

  • Definition 2.1: Standard triple
  • Theorem 2.2: Jacobson-Morozov theorem
  • Theorem 2.3
  • Corollary 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • proof
  • ...and 48 more