Speaker
Description
The Nambu-determinant Poisson brackets on R^d are expressed by the formula
{f,g}d (x) = \rho(x) \det( \partial(f,g,a_1,...a{d-2}) / \partial(x^1,...,x^d) ),
where a_1,...,a_{d-2} are smooth functions and x^1,...,x^d are global coordinates (e.g., Cartesian), so that \rho(x)\cdot\partial_{x} is the top-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 R^3.
Independently, Nambu's binary bracket {-,-}_d with Jacobian determinant and d-2 Casimirs a_1,...,a{d-2} belong to the Nambu (1973) class of N-ary multi-linear antisymmetric polyderivational brackets {-,...,-}_d which satisfy natural N-ary generalizations of the Jacobi identity for Lie algebras.
In the study of Kontsevich's infinitsimal deformations of Poisson brackets by using `good' cocycles from the graph complex, we detect case-by-case that these deformations preserve the Nambu class, and we observe new, highly nonlinear differential-polynomial identities for Jacobian determinants over affine manifolds. In this talk, several types of such identities will be presented.
(Work in progress, joint with M.Jagoe Brown, F.Schipper, and R.Buring; special thanks to the Habrok high-performance computing cluster.)
