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A Counterexample to the Quantizability of Modules

Willwacher, Thomas

In: Letters in Mathematical Physics, 2007, vol. 81, no. 3, p. 265-280

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    Title
    • A Counterexample to the Quantizability of Modules
    Author
    • Willwacher, Thomas. Department of Mathematics, ETH Zurich, Rämistrasse 101, 8092, Zurich, Switzerland
    Document Type
    • Postprint
    OAI-PMH Identifier
    • oai:doc.rero.ch:310276
    Summary
    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 $$M =\mathbb{R}^n$$ , such that: (i) The evaluation map at zero 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 still 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