Inverse Reduction for Hook-Type W-Algebras

Originating in the work of A.M. Semikhatov and D. Adamović, inverse reductions are embeddings involving W-algebras corresponding to the same Lie algebra but different nilpotent orbits. Here, we show that an inverse reduction embedding between the affine \(\mathfrak {sl}_{n+1}\) ${\mathfrak{sl}}_{n+1}$ vertex operator algebra and the minimal \(\mathfrak {sl}_{n+1}\) ${\mathfrak{sl}}_{n+1}$ W-algebra exists. This generalises the realisations for \(n=1,2\) $n=1,2$ in Adamović (Commun Math Phys 366:1025–1067, 2019), Adamović (Math Ann 1–44, 2023). A similar argument is then used to show that inverse reduction embeddings exists between all hook-type \(\mathfrak {sl}_{n+1}\) ${\mathfrak{sl}}_{n+1}$ W-algebras, which includes the principal/regular, subregular, minimal \(\mathfrak {sl}_{n+1}\) ${\mathfrak{sl}}_{n+1}$ W-algebras, and the affine \(\mathfrak {sl}_{n+1}\) ${\mathfrak{sl}}_{n+1}$ vertex operator algebra. This generalises the regular-to-subregular inverse reduction of Fehily (Commun Contemp Math 2250049, 2022), and similarly uses free-field realisations and their associated screening operators.