We propose an extension of [Formula: see text]-ary Nambu–Poisson bracket to superspace [Formula: see text] and construct by means of superdeterminant a family of Nambu–Poisson algebras of even degree functions, where the parameter of this family is an invertible transformation of Grassmann coordinates in superspace [Formula: see text]. We prove in the case of the superspaces [Formula: see text] and [Formula: see text] that our [Formula: see text]-ary bracket, defined with the help of superdeterminant, satisfies the conditions for [Formula: see text]-ary Nambu–Poisson bracket, i.e. it is totally skew-symmetric and it satisfies the Leibniz rule and the Filippov–Jacobi identity (fundamental identity). We study the structure of [Formula: see text]-ary bracket defined with the help of superdeterminant in the case of superspace [Formula: see text] and show that it is the sum of usual [Formula: see text]-ary Nambu–Poisson bracket and a new [Formula: see text]-ary bracket, which we call [Formula: see text]-bracket, where [Formula: see text] is the product of two odd degree smooth functions.
Poisson algebras via model theory and differential-algebraic geometry
