The vacuum preserving Lie algebra of a classical W-algebra

TL;DR

This paper introduces a simplified method to associate a finite Lie algebra with classical W-algebras, revealing a natural isomorphism with the original Lie algebra in the Drinfeld-Sokolov reduction context.

Contribution

It generalizes and simplifies the construction of the vacuum preserving Lie algebra for classical W-algebras, establishing a natural isomorphism with the underlying Lie algebra.

Findings

Constructed a finite Lie algebra containing the M"obius sl(2) subalgebra.

Proved the isomorphism between the vacuum preserving algebra and the original Lie algebra.

Applied the method to W-algebras from Drinfeld-Sokolov reduction.

Abstract

We simplify and generalize an argument due to Bowcock and Watts showing that one can associate a finite Lie algebra (the `classical vacuum preserving algebra') containing the M\"obius $sl(2)$ subalgebra to any classical $\W$-algebra. Our construction is based on a kinematical analysis of the Poisson brackets of quasi-primary fields. In the case of the $\W_\S^\G$-algebra constructed through the Drinfeld-Sokolov reduction based on an arbitrary $sl(2)$ subalgebra $\S$ of a simple Lie algebra $\G$, we exhibit a natural isomorphism between this finite Lie algebra and $\G$ whereby the M\"obius $sl(2)$ is identified with $\S$.