Canonical forms for the invariant tensors and A-B-C-cohomologies of integrable Hamiltonian systems

The canonical forms for the (l,m) tensors, l+m≤3, that are invariant with respect to a Liouville-integrable non-degenerate Hamiltonian system V on a symplectic manifold M2k are derived. It is proved that the characteristic polynomial of any invariant (1,1) tensor Aαβ is a perfect square; therefore its eigenvalues have even multiplicities. Any invariant metric gαβ is indefinite and has signature σ≤k. The derived canonical forms are applied to the calculation of the A-B-C-cohomologies of Liouville-integrable Hamiltonian systems