Purely non-local Hamiltonian formalism, Kohno connections and ∨-systems

In this paper, we extend purely non-local Hamiltonian formalism to a class of Riemannian F-manifolds, without assumptions on the semisimplicity of the product ? or on the flatness of the connection ▾. In the flat case, we show that the recurrence relations for the principal hierarchy can be re-interpreted using a local and purely non-local Hamiltonian operators and in this case they split into two Lenard-Magri chains, one involving the even terms, the other involving the odd terms. Furthermore, we give an elementary proof that the Kohno property and the ?-system condition are equivalent under suitable assumptions and we show how to associate a purely non-local Hamiltonian structure to any v-system, including degenerate ones