The nilpotent variety and invariant polynomial functions in the Hamiltonian algebra

TL;DR

This paper proves Premet's conjecture for Hamiltonian Lie algebras, showing their nilpotent variety is irreducible, normal, and a complete intersection, and extends the Chevalley Restriction theorem to this setting.

Contribution

It establishes the irreducibility and geometric properties of the nilpotent variety for Hamiltonian Lie algebras and generalizes key invariant theory results.

Findings

Nilpotent variety of Hamiltonian Lie algebra is irreducible

The nilpotent variety is normal and a complete intersection

Invariant polynomial ring generators are explicitly described

Abstract

Premet has conjectured that the nilpotent variety of any finite-dimensional restricted Lie algebra is an irreducible variety. In this paper, we prove this conjecture in the case of Hamiltonian Lie algebra. and show that its nilpotent variety is normal and a complete intersection. In addition, we generalize the Chevalley Restriction theorem to Hamiltonian Lie algebra. Accordingly, we give the generators of the invariant polynomial ring.