Seminarium KMMF "Teoria Dwoistości"

The Nambu-determinant Poisson brackets on $\mathbb{R}^d$ are expressedby the formula\[\{f,g\}_d (\boldsymbol{x}) = \varrho(\boldsymbol{x}) \cdot \det\bigl(\partial(f,g,a_1,...a_{d-2}) / \partial(x^1,...,x^d) \bigr),\]where $a_1$, $\ldots$, $a_{d-2}$ are smooth functions and $x^1$,$\ldots$, $x^d$ are global coordinates (e.g., Cartesian), so that$\varrho(\boldsymbol{x})\cdot\partial_{\boldsymbol{x}}$ is thetop-degree multivector. For an example of Nambu--Poisson bracket in classical mechanics,consider the Euler top with $\{x,y\}_3 = z$ and so on cyclically on$\mathbb{R}^3$. Independently, Nambu's binary bracket $\{{-},{-}\}_d$ with Jacobiandeterminant and $d-2$ Casimirs $a_1$, $\ldots$, $a_{d-2}$ belong tothe Nambu (1973) class of $N$-ary multi-linear antisymmetricpolyderivational brackets $\{{-},\ldots,{-}\}_d$ which satisfy natural$N$-ary generalizations of the Jacobi identity for Lie algebras. In the study of Kontsevich's infinitsimal deformations of Poissonbrackets by using `good' cocycles from the graph complex, we detectcase-by-case that these deformations preserve the Nambu class, and weobserve new, highly nonlinear differential-polynomial identities forJacobian determinants over affine manifolds. In this talk, severaltypes of such identities will be presented.(Work in progress, joint with M.~Jagoe Brown, F.~Schipper, andR.~Buring: see [arXiv:2112.03897] and [arXiv:2409.18875, 2409.12555,2409.15932, 2503.10916, 2503.10926]; special thanks to the Habrokhigh-performance computing cluster.)