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 ⊂ 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 W G S-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 W G S-algebras and 'free field' algebras with conformal weights ∆ ∈ {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 ∆ ≥ 0 (which is the case for all DS reductions known to date), then the ∆ ≥ 3 2 subsectors of the weights are necessarily the same as in the corresponding W G S-algebra. These results are consistent with an earlier result by Bowcock 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 ∆ ≥ 0 that may be obtained from DS reduction is exhausted by the canonical ones.