On the structure of W-algebras in type A

Thomas Creutzig, Justine Fasquel, Andrew R. Linshaw, Shigenori Nakatsuka

Abstract: We formulate and prove examples of a conjecture which describes the $\mathcal{W}$-algebras in type $A$ as successive quantum Hamiltonian reductions of affine vertex algebras associated with several hook-type nilpotent orbits. This implies that the affine coset subalgebras of hook-type $\mathcal{W}$-algebras are building blocks of the $\mathcal{W}$-algebras in type $A$. In the rational case, it turns out that the building blocks for the simple quotients are provided by the minimal series of the regular $\mathcal{W}$-algebras. In contrast, they are provided by singlet-type extensions of $\mathcal{W}$-algebras at collapsing levels which are irrational. In the latter case, several new sporadic isomorphisms between different $\mathcal{W}$-algebras are established.