In this talk, we’ll discuss in detail quantisations for two nonabelian hierarchies, namely the Volterra and the Toda hierarchies, applying a novel approach based on the notion of quantisation ideals. We prove that the nonabelian Volterra together with the whole hierarchy of its symmetries admit a deformation quantisation, and that all odd-degree symmetries of the Volterra hierarchy admit also a non-deformation quantisation. For the Toda hierarchy we find a bi-quantum structure, which can be regarded as a quantum deformation of its classical bi-Hamiltonian structure.The problem of finding the Hamiltonians of the quantised systems will also be discussed. This is a joint work with S. Carpentier and A.V. Mikhailov.
