ON THE COMPLETENESS OF THE SET OF CLASSICAL W-ALGEBRAS OBTAINED FROM DS REDUCTIONS

We clarify the notion of the DS - generalized Drinfeld-Sokolov - reduc tion approach to classical W-algebras. We first strengthen an earlier theorem which showed that an sl(2) embedding L subset-of can be associ ated 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 possi ble 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 (L)G-algebras and are called canonical, the quasi-primary basis corres ponds to the highest weights of the sl(2). Here we find some examples of noncanonical DS reductions leading to W-algebras which are direct p roducts of W(L)G-algebras and ''free field'' algebras with conformal w eights DELTA is-an-element-of 0, 1/2, 1}. We also show that if the con formal weights of the generators of a W-algebra obtained from DS reduc tion are nonnegative DELTA greater-than-or-equal-to 0 (which is the ca se for all DS reductions known to date), then the DELTA greater-than-o r-equal-to 3 subsectors of the weights are necessarily the same as in the corresponding W(L)G-algebra. These results are consistent with an earlier result by Bowcock and Watts on the spectra of W(L)G-algebras d erived by different means. We are led to the conjecture that, up to fr ee fields, the set of W-algebras with nonnegative spectra DELTA greate r-than-or-equal-to 0 that may be obtained from DS reduction is exhaust ed by the canonical ones.