Higher Poisson Brackets and Differential Forms

TL;DR

This paper extends the relationship between Poisson brackets and symplectic forms to inhomogeneous multivector fields and differential forms, introducing higher Poisson brackets and generalized symplectic structures.

Contribution

It introduces a generalized framework for symplectic structures and Poisson brackets applicable to inhomogeneous multivector fields and differential forms, including Koszul type brackets.

Findings

Generalization of symplectic structures to inhomogeneous forms

Construction of higher Poisson brackets

Development of Koszul type brackets in this setting

Abstract

We show how the relation between Poisson brackets and symplectic forms can be extended to the case of inhomogeneous multivector fields and inhomogeneous differential forms (or pseudodifferential forms). In particular we arrive at a notion which is a generalization of a symplectic structure and gives rise to higher Poisson brackets. We also obtain a construction of Koszul type brackets in this setting.