It is shown how the universal enveloping algebra of a Lie algebra $L$ can be obtained as a formal deformation of the Kirillov-Souriau Poisson algebra $C^{\infty }\left(L^{*}\right)$ of smooth functions on the dual of $L$. This deformation process may be viewed as a “quantization” in the sense of F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer [Ann. Phys. 111, 61-110 (1978; Zbl 0377.53024) and ibid., 111-151 (1978; Zbl 0377.53025)]. The result presented is a somewhat more elaborate version of earlier findings by S. Gutt [Lett. Math. Phys. 7, 249-258 (1983; Zbl 0522.58019)] and V. G. Drinfel’d [Sov. Math., Dokl. 28, 667-671 (1983); translation from Dokl. Akad. Nauk SSSR 273, No. 3, 531-535 (1983; Zbl 0553.58038)].

EUDML-ID : urn:eudml:doc:220653
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@article{701507,
     title = {Universal enveloping algebras and quantization},
     booktitle = {Proceedings of the Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {1993},
     pages = {[65]-70},
     mrnumber = {MR1246621},
     zbl = {0792.58020},
     url = {http://dml.mathdoc.fr/item/701507}
}

Grabowski, Janusz. Universal enveloping algebras and quantization, dans Proceedings of the Winter School "Geometry and Physics", GDML_Books,  (1993), pp. [65]-70. http://gdmltest.u-ga.fr/item/701507/