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

On the completeness of the set of classical W-algebras obtained from DS reductions

We clarify the notions of the DS-generalized Drinfeld-Sokolov-reduction approach to classical W-algebras and collect evidence supporting the conjecture that the canonical W-algebras (called W_S"G-algebras), defined by the highest weights of the sl(2) embeddings S is contained in G into the simple Lie algebras, essentially exhaust the set of W-algebras that may be obtained by reducing the affine Kac-Moody (KM) Poisson bracket algebras in this approach. We first prove that an sl(2) embedding S is contained in G can be associated to every DS reduction and then derive restrictions on the possible cases belonging to the same sl(2) embedding. We find examples of noncanonical DS reductions, but in all those examples the resultant noncanonical W-algebra decouples into the direct product of the corresponding W_S"G-algebra and a system of 'free fields' with conformal weights #DELTA# element of #left brace#0, 1/2, 1#right brace#. We also show that if the conformal weights of the generators of a W-algebra obtained from DS reduction are nonnegative #DELTA# #>=# 0 (which is the case for all DS reductions known to date), then the #DELTA# #>=# 3/2 subsectors of the weights are necessarily the same as in the corresponding W_S"G-algebra. The paper is concluded by a list of open problems concerning DS reductions and more general Hamiltonian KM reductions. (orig.)Available from TIB Hannover: RN 5063(93-14) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman