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

TL;DR

This paper explores the relationships between affine $ ext{W}$-algebras associated with $ ext{sl}_4$, introducing new reduction techniques and demonstrating their compatibility with existing module reductions.

Contribution

It introduces partial and inverse Hamiltonian reductions for affine $ ext{W}$-algebras in the $ ext{sl}_4$ case, filling gaps in the literature.

Findings

Constructed all missing reductions between affine $ ext{W}$-algebras

Established compatibility with Weyl module reductions

Enhanced understanding of nilpotent orbit relations

Abstract

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 $\mathfrak{sl}_4$, 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.