  The polysymplectic $(n+1)$-form is introduced as an analogue of the
symplectic form for the De Donder-Weyl polymomentum Hamiltonian formulation of
field theory. The corresponding Poisson brackets on differential forms are
constructed. The analogues of the Poisson algebra are shown to be generalized
(non-commutative and higher-order) Gerstenhaber algebras defined in the text.
