On the Completeness of the Set of Classical W-Algebras Obtained from DS Reductions

Abstract

We clarify the notion of the DS --- generalized Drinfeld-Sokolov --- reduction approach to classical $\mathcal{W}$-algebras. We first strengthen an earlier theorem which showed that an $sl(2)$ embedding $\mathcal{S}\subset \mathcal{G}$ 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 ${\mathcal{W}}_{\mathcal{S}}^{\mathcal{G}}$-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 $\W$-algebras which are direct products of ${\mathcal{W}}_{\mathcal{S}}^{\mathcal{G}}$-algebras and `free field' algebras with conformal weights $\Delta \in \{0,\frac{1}{2},1\}$. We also show that if the conformal weights of the generators of a $\mathcal{W}$-algebra obtained from DS reduction are nonnegative $\Delta \ge 0$ (which is

48 pages, plain TeX, BONN-HE-93-14, DIAS-STP-93-02

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