The Z2-graded Schouten–Nijenhuis bracket and generalized super-Poisson structures

The super or Z2-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 Λ of even order. The corresponding super “Jacobi identities” are expressed by stating that these tensors have a zero super Schouten–Nijenhuis bracket with themselves [Λ,Λ]=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 representative example.

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