On the classification of quantum W-algebras

Details On the classification of quantum W-algebras On the classification of quantum W-algebras

1991-11-28

arxiv.org/abs/hep-th/9111062v1

Nucl.Phys. B379 (1992) 63-95

Physics

High Energy Physics

High Energy Physics - Theory

28 pages

Scientific paper

10.1016/0550-3213(92)90590-8

In this paper we consider the structure of general quantum W-algebras. We introduce the notions of deformability, positive-definiteness, and reductivity of a W-algebra. We show that one can associate a reductive finite Lie algebra to each reductive W-algebra. The finite Lie algebra is also endowed with a preferred $sl(2)$ subalgebra, which gives the conformal weights of the W-algebra. We extend this to cover W-algebras containing both bosonic and fermionic fields, and illustrate our ideas with the Poisson bracket algebras of generalised Drinfeld-Sokolov Hamiltonian systems. We then discuss the possibilities of classifying deformable W-algebras which fall outside this class in the context of automorphisms of Lie algebras. In conclusion we list the cases in which the W-algebra has no weight one fields, and further, those in which it has only one weight two field.

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