Building blocks of W-algebras and duality

W-algebras are a family of vertex algebras obtained as Hamiltonian reductions of affine vertex algebras parametrized by nilpotent orbits.The W-algebras associated with regular nilpotent orbits enjoy the Feigin-Frenkel duality. More recently, Gaiotto and Rap\v{c}\'ak generalize this result to hook-type W-algebras as the triality for vertex algebras at the corner.
In this talk, I will present the correspondence of representation categories for the hook-type W-superalgebras and how to gain general W-algebras in type A from hook-type W-algebras. The talk is based on joint works with Creutzig, Fasquel, Genra, Linshaw, and Sato.