zbMATH — the first resource for mathematics

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].

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