The Z_2-graded Schouten-Nijenhuis bracket and generalized super-Poisson structures

TL;DR

This paper introduces a super Z_2-graded Schouten-Nijenhuis bracket and uses it to define new generalized super-Poisson structures, expanding the mathematical framework for superalgebraic structures with potential applications in theoretical physics.

Contribution

It develops a super Z_2-graded Schouten-Nijenhuis bracket and constructs new generalized super-Poisson structures based on graded-skew-symmetric tensors, including explicit examples on dual spaces of superalgebras.

Findings

Defined the super Z_2-graded Schouten-Nijenhuis bracket.

Constructed new generalized super-Poisson structures.

Provided explicit examples on dual spaces of superalgebras.

Abstract

The super or Z_2-graded Schouten-Nijenhuis bracket is introduced. Using it, new generalized super-Poisson structures are found which are given in terms of certain graded-skew-symmetric contravariant tensors \Lambda of even order. The corresponding super `Jacobi identities' are expressed by stating that these tensors have zero super Schouten-Nijenhuis bracket with themselves [\Lambda,\Lambda]=0. As a particular case, we provide the linear generalized super-Poisson structures which can be constructed on the dual spaces of simple superalgebras with a non-degenerate Killing metric. The su(3,1) superalgebra is given as a generic example.