THE VACUUM PRESERVING LIE-ALGEBRA OF A CLASSICAL W-ALGEBRA

We simplify and generalize an argument due to Bowcock and Watts showin g that one can associate a finite Lie algebra (the ''classical vacuum preserving algebra'') containing the Mobius sl (2) subalgebra to any c lassical W-algebra. Our construction is based on a kinematical analysi s 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 exhi bit a natural isomorphism between this finite Lie algebra and g whereb y the Mobius sl(2) is identified with S.