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 ${\cal W}$-algebras. We first strengthen an earlier theorem which showed that an $sl(2)$ embedding ${\cal S} \subset {\cal G}$ can be associated to every DS reduction. We then use the fact that a $\cal W$-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 $\cal W$-algebras are denoted by $\cal W_S^G$-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 $\cal W$-algebras which are direct products of $\cal W_S^G$-algebras and \lq\lq free field'' algebras with conformal weights $\Delta \in \{0,{1 \over 2},1\}$. We also show that if the conformal weights of the generators of a $\cal W$-algebra obtained from DS reduction are nonnegative $\Delta \geq 0$ (which is the case for all DS reductions known to date), then the $\Delta \geq {3 \over 2}$ subsectors of the weights are necessarily the same as in the corresponding $\cal W_S^G$-algebra. These results are consistent with an earlier result by Bowcock and Watts on the spectra of $\cal W$-algebras derived by different means. We are led to the conjecture that, up to free fields, the set of $\cal W$-algebras with nonnegative spectra $\Delta \geq 0$ that may be obtained from DS reduction is exhausted by the canonical ones