On the classification of quantum W-algebras

TL;DR

This paper develops a classification framework for quantum W-algebras by introducing key structural notions and associating them with reductive Lie algebras, extending to mixed bosonic and fermionic cases, and exploring automorphisms.

Contribution

It introduces the concepts of deformability, positive-definiteness, and reductivity for quantum W-algebras, and establishes a link between reductive W-algebras and reductive Lie algebras with an $sl(2)$ subalgebra.

Findings

Associates reductive Lie algebras with reductive W-algebras.

Extends classification to W-algebras with bosonic and fermionic fields.

Lists cases with no weight one fields and only one weight two field.

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.