Towards the classification of homogeneous third-order Hamiltonian operators

Let V be a vector space of dimension n + 1. We demonstrate that n-component third-order Hamiltonian operators of differential-geometric type are parametrised by the algebraic variety of elements of rank n in S2(Λ2V) that lie in the kernel of the natural map S2(Λ2V)→Λ4V. Non-equivalent operators correspond to different orbits of the natural action of SL(n + 1). Based on this result, we obtain a classification of such operators for n≤4.