Traces for star products on the dual of a Lie algebra

Details Traces for star products on the dual of a Lie algebra Traces for star products on the dual of a Lie algebra

2002-02-13

arxiv.org/abs/math/0202126v2

Mathematics

Quantum Algebra

18 pages, LaTeX2e. Updated references

Scientific paper

10.1142/S0129055X03001643

In this paper, 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.

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