There is a well-known relationship between finite W-algebras and Yangians. The work of Rogoucy and Sorba on the "rectangular case" in type A eventually led Brundan and Kleshchev to introduce shifted Yangians, which surject onto the finite W-algebras for general linear Lie algebras. Thus, these W-algebras can be realised as truncated shifted Yangians. In parallel, the work of Ragoucy and then Brown showed that truncated twisted Yangians are isomorphic to the finite W-algebra associated to a rectangular nilpotent element in a Lie algebra of type B, C or D. For many years there has been a hope that this relationship can be extended to other nilpotent elements.
I will report on a joint work with Lewis Topley in which we introduced the shifted twisted Yangians, following the work of Lu-Wang-Zhang, and described Poisson isomorphisms between their truncated semiclassical degenerations and the functions Slodowy slices associated with even nilpotent elements in classical simple Lie algebras( which can be viewed as semiclassical W-algebras). I will also mention a work in progress with Lu-Peng-Topley-Wang which deals with the quantum analogue of our theorem.
I will also recall what Poisson algebras and (filtered) quantizations are and give a brief intro to Slodowy slices, finite W-algebras and Yangians so that the talk should be quite accessible.