### Abstract

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 L {square subset}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 give sl(2) embedding. In the known DS reductions found to data, for which the W-algebras are denoted by W_{L}^{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 W_{L}^{G}-algebras and "free field" algebras with conformal weights Δ∈{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 W_{L}^{G}-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 Δ>-0 that may be obtained from DS reduction is exhausted by the canonical ones.

Original language | English |
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Pages (from-to) | 399-431 |

Number of pages | 33 |

Journal | Communications in Mathematical Physics |

Volume | 162 |

Issue number | 2 |

DOIs | |

Publication status | Published - May 1 1994 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*162*(2), 399-431. https://doi.org/10.1007/BF02102024