An explicit *-product on the cotangent bundle of a Lie group.(English)Zbl 0522.58019

Summary: We give explicit formulas for a *-product on the cotangent bundle $$T^*G$$ of a Lie group $$G$$; these formulas involve on the one hand the multiplicative structure of the universal enveloping algebra $$U(\mathfrak g)$$ of the Lie algebra $$\mathfrak g$$ of $$G$$ and on the other hand bidifferential operators analogous to the ones used by Moyal to define a *-product on $$\mathbb R^{2n}$$.

MSC:

 53D55 Deformation quantization, star products 53B50 Applications of local differential geometry to the sciences 22E46 Semisimple Lie groups and their representations
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References:

 [1] Bateman, H., Higher Transcendental Functions (1953) p. 38. · Zbl 0051.34703 [2] BayenF., FlatoM., FronsdalC., LichnerowiczA., and SternheimerD., Ann. Phys. 111, 61-151 (1978). · Zbl 0377.53024 [3] CahenM. and GuttS., Lett. Math. Phys. 6, 395-404 (1982). · Zbl 0522.58018 [4] Dixmier, J., Universal Enveloping Algebra (1978) p. 96. [5] GerstenhaberM., Ann. Math. 79, 59-103 (1964). · Zbl 0123.03101 [6] LichnerowiczA., Lett. Math. Phys. 2, 133-143 (1977). · Zbl 0392.58019 [7] MoyalJ., Proc. Cambridge Phil. Soc. 45, 99-124 (1949). [8] VeyJ., Comm. Math. Helv. 50, 421-454 (1975). · Zbl 0351.53029
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