Purely non-local Hamiltonian formalism, Kohno connections and $\vee$-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 $\circ$ or on the flatness of the connection $\nabla$. 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 $\vee$-system condition are equivalent under suitable conditions and we show how to associate a purely non-local Hamiltonian structure to any $\vee$-system, including degenerate ones.