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

Nambu-determinant brackets on $R^d\ni x=(x^1,\mathrm{.}\mathrm{.}\mathrm{.},x^d)$, $\{f,g{\}}_d(x)=\rho (x)\det (\mathrm{\partial }(f,g,a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_{d-2})\mathrm{/}\mathrm{\partial }(x^1,\mathrm{.}\mathrm{.}\mathrm{.},x^d))$, with $a_i\in C^{\mathrm{\infty }}(R^d)$ and $\rho {\mathrm{\partial }}_x\in {\mathfrak{X}}^d(R^d)$, are a class of 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 bivectors $P$ on affine $R_{aff}^d$ – elements $\dot{P}=Q^{\gamma }(P)\in H_P^2(R_{aff}^d)$ in the Lichnerowicz-Poisson second cohomology groups; we note that known graph cocycles $\gamma$ preserve the Nambu-Poisson class $\{P(\rho ,a)\}$, and we express, directly from $\gamma$, the evolution $\dot{\rho }$,$\dot{a}$ that induces $\dot{P}$. Over all $d\ge 2$ at once, there is no universal mechanism for the bivector cocycles $Q_d^{\gamma }$ to be trivial, $Q_d^{\gamma }=[\text{\hspace{0.17em}⁣}[P,{\vec{X}}_d^{\gamma }(P)]\text{\hspace{0.17em}⁣}]$, w.r.t. vector fields defined uniformly for all dimensions $d$ by the same graph formula. While over $R^2$, the graph flows $\dot{P}=Q_{2D}^{{\gamma }_i}(P(\rho ))$ for $\gamma \in \{{\gamma }_3,{\gamma }_5,{\gamma }_7,\mathrm{.}\mathrm{.}\mathrm{.}\}$ are trivialized by vector fields ${\vec{X}}_{2D}^{{\gamma }_i}=(dx\wedge dy{)}^{-1}{d}_{dR}(Ha{m}^{{\gamma }_i}(P))$ of peculiar shape, we detect that in $d\ge 3$, the 1-vectors from 2D, now with $P(\rho ,a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_{d-2})$ inside, do not solve the problems $Q_{d\ge 3}^{{\gamma }_i}=[\text{\hspace{0.17em}⁣}[P,{\vec{X}}_{d\ge 3}^{{\gamma }_i}(P(\rho ,a))]\text{\hspace{0.17em}⁣}]$, yet they do yield good Ansatz where we find solutions ${\vec{X}}_{d=3,4}^{{\gamma }_i}(P(\rho ,a))$. In the study of the step $d\mapsto d+1$, by adapting the Kontsevich graph calculus to the Nambu-Poisson class of brackets, we discover more identities for the Jacobian determinants within $P(\rho ,a)$, i.e. for multivector-valued $GL(d)$-invariants on $R_{aff}^d$.