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Tangential deformation quantization and polarized symplectic groupoids. (English) Zbl 1166.53324
Sternheimer, Daniel (ed.) et al., Deformation theory and symplectic geometry. Proceedings of the Ascona meeting, Switzerland, June 17–21, 1996. Dordrecht: Kluwer Academic Publishers (ISBN 0-7923-4525-8/hbk). Math. Phys. Stud. 20, 301-314 (1997).
Summary: We derive geometric analogues of theorems of Cahen-Gutt-Rawnsley and Asin Lares on deformation quantizations compatible with coadjoint orbits. The symmetrization isomorphism between the universal enveloping algebra \(U(\mathfrak{g})\) and the symmetric algebra \(S(\mathfrak{g})\) can be used [F. A. Berezin, Funct. Anal. Appl. 1, 91–102 (1967); translation from Funkts. Anal. Prilozh. 1, No. 2, 1-14 (1967; Zbl 0227.22020), S. Gutt, Lett. Math. Phys. 7, 249–258 (1983; Zbl 0522.58019)] to produce a “standard” deformation quantization F. Bayen, M. Flato, C. Frønsdal, A. Lichnerowicz and D. Sternheimer[Ann. Phys. 111, 61–110, 111–151 (1978; Zbl 0377.53024, Zbl 0377.53025)] for the Lie-Poisson structure on the dual \(\mathfrak{g}^*\) of the Lie algebra \(\mathfrak{g}\). Since the symplectic leaves of \(\mathfrak{g}^*\) are the coadjoint orbits, it is interesting to know whether this, or any, deformation quantization of \(\mathfrak{g}^*\) restricts to give deformation quantizations of the orbits. Cahen, Gutt, and Rawnsley (loc. cit) have shown that a deformation quantization of \(\mathfrak{g}^*\) by bidifferential operators compatible with the coadjoint orbit decomposition can exist only if \(\mathfrak{g}\) satisfies a very strong algebraic condition. No semisimple Lie algebras satisfy this condition. In addition, S. A. Lares [J. Geom. Phys. 24, No. 2, 164–172(1997; Zbl 0891.17019)] has shown that the standard deformation quantization of \(\mathfrak{g}^*\) restricts to a given coadjoint orbit \(\mathcal{O}\) if and only if \(\mathcal{O}\) is an open subset of an affine subspace in \(\mathfrak{g}^*\).
For the entire collection see [Zbl 0923.00023].

MSC:
53D55 Deformation quantization, star products
58H05 Pseudogroups and differentiable groupoids
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