Automorphisms of Liouville Structures

TL;DR

This paper characterizes all automorphisms of Liouville structures on symplectic vector spaces, especially when the potential differs from the canonical form by a homogeneous monomial's differential.

Contribution

It provides a complete description of automorphisms for Liouville structures with potentials differing by homogeneous monomials.

Findings

All automorphisms are explicitly determined for the specified class.

The structure of automorphisms depends on the form of the potential.

The results clarify symmetries of Liouville structures in symplectic vector spaces.

Abstract

By a Liouville structure on a symplectic manifold $(M, \omega)$ we mean a choice of symplectic potential: that is, a choice of one-form $\theta$ on $M$ such that ${\rm d} \theta = \omega$. We determine precisely all the automorphisms of a Liouville structure in case $(M, \omega)$ is a symplectic vector space and $\theta$ differs from its canonical symplectic potential by the differential of a homogeneous monomial.