Nambu tensors and commuting vector fields

Takhtajan has recently studied the consistency conditions for Nambu brackets, and suggested that they have to be skew-symmetric, and satisfy Leibnitz rule and the Fundamental Identity (FI, it is a generalization of the Jacobi identity). If the n-th order Nambu brackets in dimension N is written as {f1, . . . , fn} = ηi1...in∂i1f1 · · · ∂infn (where the iα summations range over 1 . . . N), the FI implies two conditions on the Nambu tensor η, one algebraic and one differential. The algebraic part of FI implies decomposability of η and in this letter we show that the Nambu bracket can then be written as {f1, . . . , fn} = ρ ǫα1...αnD̄ 1f1 · · · D̄ fn, where ǫα1...αn is the usual totally antisymmetric n-dimensional tensor, the αi summations range over 1 . . . n, and D̄ := ∂α+ ∑N k=n+1 v α k ∂k are n vector fields. Our main result is that the differential part of the FI is satisfied iff the vector fields D̄ commute. Examples are provided by integrable Hamiltonian systems. It turns out that then the Nambu bracket itself guarantees that the motions stays on the manifold defined by the constants of motion of the integrable system, while the n − 1 Nambu Hamiltonians determine the (possibly non-integrable) motion on this manifold.