A New Look at the Schouten-Nijenhuis, Fr\"olicher-Nijenhuis and Nijenhuis-Richardson Brackets for Symplectic Spaces

TL;DR

This paper re-expresses key algebraic brackets on symplectic spaces using extended Poisson brackets from classical mechanics, offering a new perspective on their structure and relationships.

Contribution

It introduces a novel formulation of the Schouten-Nijenhuis, Fr"olicher-Nijenhuis, and Nijenhuis-Richardson brackets via extended Poisson brackets in symplectic geometry.

Findings

Unified framework for brackets using extended Poisson structures

Simplifies calculations in symplectic geometry

Provides insights into classical mechanics formulations

Abstract

In this paper we re-express the Schouten-Nijenhuis, the Fr\"olicher-Nijenhuis and the Nijenhuis-Richardson brackets on a symplectic space using the extended Poisson brackets structure present in the path-integral formulation of classical mechanics.