On field theoretic generalizations of a Poisson algebra

A few generalizations of a Poisson algebra to field theory canonically formulated in terms of the polymomentum variables are discussed. A graded Poisson bracket on differential forms and an (n + l)-ary bracket on functions are considered. The Poisson bracket on differential forms gives rise to various generalizations of a Gerstenhaber algebra: the noncommutative (in the sense of Loday) and the higher-order (in the sense of the higher-order graded Leibniz rule). The (n + l)-ary bracket fulfills the properties of the Nambu bracket including the “fundamental identity”, thus leading to the Nambu-Poisson algebra. We point out that in the field theory context the Nambu bracket with a properly defined covariant analogue of Hamiltonʹs function determines a joint evolution of several dynamical variables.