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Traces for star products on the dual of a Lie algebra. (English) Zbl 1129.53305

Summary: We describe all traces for the BCH star-product on the dual of a Lie algebra. First we show by an elementary argument that the BCH as well as the Kontsevich star-product are strongly closed if and only if the Lie algebra is unimodular. In a next step we show that the traces of the BCH star-product are given by the ad-invariant functionals. Particular examples are the integration over coadjoint orbits. We show that for a compact Lie group and a regular orbit one can even achieve that this integration becomes a positive trace functional. In this case we explicitly describe the corresponding GNS representation. Finally we discuss how invariant deformations on a group can be used to induce deformations of spaces where the group acts on.

MSC:

53D55 Deformation quantization, star products
17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B63 Poisson algebras
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