Cohomologies of the Poisson superalgebra

Abstract

We consider antiPoisson superalgebras realized on the smooth Grassmann-valued functions with compact support in R n and with the grading inverse to Grassmanian parity. The lower cohomologies of these superalgebras are found. IntroductionThe odd Poisson bracket play an important role in Lagrangian formulation of the quantum theory of the gauge fields, which is known as BV-formalism The antibracket possesses many features analogous to those of the Poisson bracket and even can be obtained via "canonical formalism" with an "odd time". However, unlike the Poisson bracket, on different aspects of whose deformations (quantization) there is voluminous literature, the deformations of the antibracket is not satisfactorily studied yet. The only result is [7], where the deformations of the Poisson and antibracket superalgebras realized on the superspace of polynomials are found.The goal of present work is finding the lower cohomology spaces of antiPoisson superalgebra realized on the smooth Grassmann-valued functions with compact support in R n . These results is used in the next work [8] where the general form of the deformation of such antiPoisson superalgebra is found. Particularly, it is shown in [8] that the nontrivial deformations do exist.Let K be either R or C. We denote by D(R n ) the space of smooth K-valued functions with compact support on R n . This space is endowed with its standard topology. We set

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