The Schouten-Nijenhuis bracket, cohomology and generalized Poisson structures

TL;DR

This paper explores generalized Poisson structures defined by skew-symmetric tensors, analyzing their properties through the Schouten-Nijenhuis bracket and cohomology, and classifies structures on duals of simple Lie algebras.

Contribution

It introduces a framework for generalized Poisson structures using the Schouten-Nijenhuis bracket and characterizes structures on duals of simple Lie algebras.

Findings

Conditions for generalized Poisson structures via cohomology

Classification of linear structures on duals of simple Lie algebras

Expression of Jacobi identities in terms of tensors

Abstract

Newly introduced generalized Poisson structures based on suitable skew-symmetric contravariant tensors of even order are discussed in terms of the Schouten-Nijenhuis bracket. The associated `Jacobi identities' are expressed as conditions on these tensors, the cohomological contents of which is given. In particular, we determine the linear generalized Poisson structures which can be constructed on the dual spaces of simple Lie algebras.