Quantum BRST Charge for Quadratically Nonlinear Lie Algebras

We consider the construction of a nilpotent BRST charge for extensions of the Virasoro algebra of the form {Ta, Tb} =f,{Tc+ V~,~TcTd, (classical algebras in terms of Poisson brackets) and [To, TbJ = habI + f~{ T~ + V~ (T~ Te) (quantum algebras in terms of commutator brackets; normal ordering of the product (T~ Te) is understood). In both cases we assume that the set of generators {To} splits into a set {Hi} generating an ordinary Lie algebra and remaining generators {S,}, such that only the V~ are nonvanishing. In the classical case a nilpotent BRST charge can always be constructed; for the quantum case we derive a condition which is necessary and sufficient for the existence of a nilpotent BRST charge. Non-trivial examples are the spin-3 algebra with central charge c = 100 and the so(N)-extended superconformal algebras with level S = 2(N 3).