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

We clarify the notion of the DS - generalized Drinfeld-Sokolov - reduction approach to classical W-algebras. We first strengthen an earlier theorem which showed that an sl(2) embedding S contains or equal to G can be associated to every DS reduction. We then use the fact that a W-algebra must have a quasi-primary basis to derive severe restrictions on the possible reductions corresponding to a given sl(2) embedding. In the known DS reductions found to date, for which the W-algebras are denoted by WSG-algebras and are called canonical, the quasi-primary basis corresponds to the highest weights of the sl(2). Here we find some examples of noncanonical DS reductions leading to W-algebras which are direct products of WSG-algebras and ''free field'' algebras with conformal weights Δ element of {0,1/2,1}. We also show that if the conformal weights of the generators of a W-algebra obtained from DS reduction are nonnegative Δ≥0 (which is the case for all DS reductions known to date), then the Δ≥3/2 subsectors of the weights are necessarily the same as in the corresponding WSG-algebra. These results are consistent with an earlier result by Browcock and Watts on the spectra of W-algebras derived by different means. We are led to the conjecture that, up to free fields, the set of W-algebras with nonnegative spectra Δ≥0 that may be obtained from DS reduction is exhausted by the canonical ones. (orig.)