NAMBU TENSORS AND COMMUTING VECTOR FIELDS

Takhtajan has recently studied the consistency conditions for Nambu brackets (1994 Commun. Math. Phys. 160 295 - 315), and suggested that they have to be skew-symmetric, and satisfy the Leibnitz rule and the fundamental identity (FI, a generalization of the Jacobi identity). If the nth-order Nambu brackets in dimension N are written as (where the summations range over ), the FI implies two conditions on the Nambu tensor , one algebraic and one differential. The algebraic part of FI implies decomposability of and in this letter we show that the Nambu bracket can then be written as , where is the usual totally antisymmetric n-dimensional tensor, the summations range over , and are n vector fields. Our main result is that the differential part of the FI is satisfied iff the vector fields 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.