We will present a new approach to the problem of quantising integrable systems of differential-difference equations. The main idea is to lift these systems to systems defined on free associative algebras and look for the ideals there that are stabilized by the dynamics of the lifted systems. In a reasonable class of candidate ideals, there are typically very few that are invariant for the first equation in the integrable hierarchy. Once these ideals are picked the challenge is to prove that the whole hierarchy preserves them. We will discuss these ideas using as a key example the hierarchy of the Bogoyavlensky equation. This is a joint work with A. Mikhailov (Leeds) and J. P. Wang (U. of Kent). It is soon to be published but one can already find an introduction here : arXiv:2009.01838
Thomas Strobl