A convenient criterion under which Z_2-graded operators are Hamiltonian

TL;DR

This paper introduces a straightforward criterion for identifying when Z_2-graded skew-adjoint differential operators are Hamiltonian, based on their closure properties in Lie algebras of evolutionary vector fields.

Contribution

It provides a new, simple criterion for Hamiltonian operators in the context of Z_2-graded differential operators, expanding the tools for analyzing such structures.

Findings

Criterion simplifies the identification of Hamiltonian operators.

Applicable to operators with images closed under commutation.

Enhances understanding of graded Hamiltonian structures.

Abstract

We formulate a simple and convenient criterion under which skew-adjoint Z_2-graded total differential operators are Hamiltonian, provided that their images are closed under commutation in the Lie algebras of evolutionary vector fields on the infinite jet spaces for vector bundles over smooth manifolds.