Z-graded extensions of Poisson brackets

A Z-graded Lie bracket { , }P on the exterior algebra Ω(M) of differential forms, which is an extension of the Poisson bracket of functions on a Poisson manifold (M,P ), is found. This bracket is simultaneously graded skew-symmetric and satisfies the graded Jacobi identity. It is a kind of an ‘integral’ of the Koszul-Schouten bracket [ , ]P of differential forms in the sense that the exterior derivative is a bracket homomorphism: [dμ, dν]P = d{μ, ν}P . A naturally defined generalized Hamiltonian map is proved to be a homomorphism between { , }P and the Frölicher-Nijenhuis bracket of vector valued forms. Also relations of this graded Poisson bracket to the Schouten-Nijenhuis bracket and an extension of { , }P to a graded bracket on certain multivector fields, being an ‘integral’ of the Schouten-Nijenhuis bracket, are studied. All these constructions are generalized to tensor fields associated with an arbitrary Lie algebroid.