On the classification of quantum W-algebras

TL;DR

The paper develops a general framework for classifying and understanding quantum W-algebras by introducing deformability, positive-definiteness, and reductivity. It proves that reductive W-algebras associate to a reductive finite Lie algebra $g$ endowed with an $sl(2)$ embedding, which fixes conformal weights and extends to bosonic/fermionic content via generalized Drinfeld–Sokolov reductions. A central construction is the vacuum-preserving algebra (vpa); its linearised version yields a finite Lie algebra that, for positive-definite classical limits, decomposes as a semisimple plus abelian algebra, i.e., a reductive structure. The work connects W-algebras to finite Lie algebras through Drinfeld–Sokolov data, provides a concrete method to obtain classical W-algebras for given $(g,sl(2))$, and offers a program to classify deformable W-algebras, including automorphism-induced families and those without Kac–Moody components. This framework clarifies how the field content and spins arise from $sl(2)$ embeddings and lays groundwork for future quantum classifications and uniqueness analyses.

Abstract

In this paper we consider the structure of general quantum W-algebras. We introduce the notions of deformability, positive-definiteness, and reductivity of a W-algebra. We show that one can associate a reductive finite Lie algebra to each reductive W-algebra. The finite Lie algebra is also endowed with a preferred $sl(2)$ subalgebra, which gives the conformal weights of the W-algebra. We extend this to cover W-algebras containing both bosonic and fermionic fields, and illustrate our ideas with the Poisson bracket algebras of generalised Drinfeld-Sokolov Hamiltonian systems. We then discuss the possibilities of classifying deformable W-algebras which fall outside this class in the context of automorphisms of Lie algebras. In conclusion we list the cases in which the W-algebra has no weight one fields, and further, those in which it has only one weight two field.