The vacuum preserving Lie algebra of a classical W-algebra

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 $\S$.Comment: 11 pages, BONN-HE-93-25, DIAS-STP-93-13. Some typos had been removed, no change in formula