Kontsevich graphs act on Nambu-Poisson brackets, I. New identities for Jacobian determinants

Nambu-determinant brackets on ℝ^d 3 x = (x¹,..., x^d), {f, g}_d(x) = %(x) · det(∂(f, g, a₁,..., a_d−2)/∂(x¹,..., x^d)⁾, with a_i ∈ C^∞(ℝ^d) and % · ∂_x ∈ X^d(ℝ^d), are a class of degenerate (rank ≤ 2) Poisson structures with (non)linear coefficients, e.g., polynomials of arbitrarily high degree. With ‘good’ cocycles in the graph complex, Kontsevich associated universal - for all Poisson bi-vectors P on affine ℝ^d_aff - elements Ṗ = Q^γ([P]) ∈ H²_P(ℝ^d_aff⁾ in the Lichnerowicz-Poisson second cohomology groups; we note that known graph cocycles γ preserve the Nambu-Poisson class^{P(%, [a])^}, and we express, directly from γ, the evolution %̇, ȧ that induces Ṗ. Over all d ≥ 2 at once, there is no ‘universal’ mechanism for the bi-vector cocycles Q^γ_d to be trivial, Q^γ_d = [[P, X̃_d^γ([P])]], w.r.t. vector fields defined uniformly for all dimensions d by the same graph formula. While over ℝ², the graph flows Ṗ = Q^γ_2Dⁱ(P(%)⁾ for γ ∈ {γ₃, γ₅, γ₇,...} are trivialised by vector fields X̃₂^γ_Dⁱ = (dx ∧ dy)⁻¹d_dR(Ham^γi(P)⁾ of peculiar shape, we detect that in d ≥ 3, the 1-vectors from 2D, now with P(%,a₁,...,a_d−2) inside, do not solve the problems Q^γ_d>ⁱ₃ = [[P, X̃_d^γ_>ⁱ₃(P(%, [a]))]], yet they do yield a good Ansatz where we find solutions X̃_d^γ₌₃ⁱ₄(P(%, [a])⁾. In the study of the step d 7→ d + 1, by adapting the Kontsevich graph calculus to the Nambu-Poisson class of brackets, we discover more identities for the Jacobian determinants within P(%, [a]), i.e.