A Counterexample to the Quantizability of Modules

Abstract

Let a Poisson structure on a manifold M be given. If it vanishes at a point m, the evaluation at m defines a one dimensional representation of the Poisson algebra of functions on M. We show that this representation can, in general, not be quantized. Precisely, we give a counterexample for M=R^n, such that: (i) The evaluation map at 0 can not be quantized to a representation of the algebra of functions with product the Kontsevich product associated to the Poisson structure. (ii) For any formal Poisson structure extending the given one and vanishing at zero up to second order in epsilon, (i) still holds. We do not know whether the second claim remains true if one allows the higher order terms in epsilon to attain nonzero values at zero.