The Koszul-Schouten bracket, i.e. a Poisson bracket extended to all differential forms \(\Omega (M)\), is described by means of graded derivations on \(\Omega (M)\). The analogous problem for graded symplectic forms is studied. In the non-degenerate case, the Koszyl-Schouten bracket comes from a graded symplectic exact form. On the other hand it is shown that any odd symplectic form is the image of a graded symplectic form on \(\Omega (M)\) of \(\mathbb{Z}\)-degree 1 by the pullback of an automorphism of the algebra \(\Omega (M)\).
For applications see also [J. Monterde, Differ. Geom. Appl. 2, No. 1, 81-97 (1992; Zbl 0789.53020)], [A. P. Nersessian, JETP Lett. 58, No. 1, 66-70 (1993); translation from Puma Zh. Eksper. Teoret. Fiz. 58, No. 1, 64-68 (1993)]and [A. Schwarz, Commun. Math. Phys. 155, No. 2, 249-260 (1993; Zbl 0786.58017)].