W-algebras are vertex algebras obtained form quantum Hamiltonian reductions of an affine vertex algebra. These reductions are naturally upgraded to functors from the category of modules over the affine vertex algebras to the category of the modules over the corresponding W-algebras. However, they are difficult to control in general.
Recently, two approaches have been developed to improve our understanding of the functors. One consists in spitting the functor into small pieces that are easier to deal with (partial reductions), the other aims to reverse the quantum Hamiltonian reduction procedure (inverse Hamiltonian reductions).
In this talk, I will discuss about recent advances in these complementary approaches.
The talk is based on recent papers with T. Creutzig, A. Linshaw and S. Nakatsuka and with Z. Fehily, E. Fursman and S. Nakatsuka
