Connecting affine $\mathcal{W}$-algebras: A case study on ${\mathfrak{s}\mathfrak{l}}_4$

We introduce the partial reductions and inverse Hamiltonian reductions between affine $\mathcal{W}$-algebras along the closure relations of associated nilpotent orbits in the case of ${\mathfrak{s}\mathfrak{l}}_4$, fulfilling all the missing constructions in the literature. We also apply the partial reductions to modules in the Kazhdan-Lusztig category and show compatibility with the usual reductions of Weyl modules.