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On the existence of star products on quotient spaces of linear Hamiltonian torus actions. (English) Zbl 1184.53087

The purpose of this note is to discuss BFV deformation quantization [M. Bordemann, H.-C. Herbig and M. J. Pflaum, in: Singularity theory. Proceedings of the 2005 Marseille singularity school and conference, CIRM, Marseille, France, January 24–February 25, 2005. Dedicated to Jean-Paul Brasselet on his 60th birthday. Singapore: World Scientific. 443–461 (2007; Zbl 1128.53060)] in the special case of a linear Hamiltonian torus action.
In particular it is shown that the Koszul complex on the moment map of an effective linear Hamiltonian torus action is acyclic. The authors rephrase the nonpositivity condition of J. M. Arms, M. J. Gotay and G. Jennings [Adv. Math. 79, No. 1, 43–103 (1990; Zbl 0721.53033)] for linear Hamiltonian torus actions. Finally, by BFV-machinery, it follows that reduced spaces of such actions admit continuous star products.

MSC:

53D55 Deformation quantization, star products
53D20 Momentum maps; symplectic reduction
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[1] Arms, J.M., Cushman, R.H., Gotay, M.J.: A Universal Reduction Procedure for Hamiltonian Group Actions, The Geometry of Hamiltonian Systems (Berkeley, CA, 1989). Math. Sci. Res. Inst. Publ., vol. 22, pp. 33–51. Springer, New York (1991) · Zbl 0742.58016
[2] Arms J.M., Gotay M.J., Jennings G.: Geometric and algebraic reduction for singular momentum maps. Adv. Math. 79(1), 43–103 (1990) · Zbl 0721.53033 · doi:10.1016/0001-8708(90)90058-U
[3] Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D.: Deformation theory and quantization. I. Deformations of symplectic structures. Ann. Phys. 111(1), 61–110 (1978) · Zbl 0377.53024 · doi:10.1016/0003-4916(78)90224-5
[4] Bierstone E., Milman P.D.: Geometric and differential properties of subanalytic sets. Ann. Math. (2) 147(3), 731–785 (1998) · Zbl 0912.32006 · doi:10.2307/120964
[5] Bierstone E., Schwarz G.W.: Continuous linear division and extension of \({\mathcal{C}^{\infty}}\) functions. Duke Math. J. 50(1), 233–271 (1983) · Zbl 0521.32008 · doi:10.1215/S0012-7094-83-05011-1
[6] Bordemann, M., Herbig, H.-C., Pflaum, M.J.: A Homological Approach to Singular Reduction in Deformation Quantization, Singularity Theory, pp. 443–461. World Scientific, Hackensack (2007) · Zbl 1128.53060
[7] Bordemann M., Herbig H.-C., Waldmann S.: BRST cohomology and phase space reduction in deformation quantization. Commun. Math. Phys. 210(1), 107–144 (2000) · Zbl 0961.53046 · doi:10.1007/s002200050774
[8] Bosio F., Meersseman L.: Real quadrics in C n , complex manifolds and convex polytopes. Acta Math. 197(1), 53–127 (2006) · Zbl 1157.14313 · doi:10.1007/s11511-006-0008-2
[9] de Medrano S.L.: Topology of the Intersection of Quadrics in R n , Algebraic Topology (Arcata, CA, 1986). Lecture Notes in Mathematics, vol. 1370, pp. 280–292. Springer, Berlin (1989)
[10] Félix Y., Oprea J., Tanré D.: Algebraic Models in Geometry, Oxford Graduate Texts in Mathematics, vol. 17. Oxford University Press, Oxford (2008) · Zbl 1149.53002
[11] Guillemin, V., Ginzburg, V., Karshon, Y.: Moment Maps, Cobordisms, and Hamiltonian Group Actions. Mathematical Surveys and Monographs, vol. 98. American Mathematical Society, Providence (2002) (Appendix J by Maxim Braverman) · Zbl 1197.53002
[12] Henneaux M., Teitelboim C.: Quantization of Gauge Systems. Princeton University Press, Princeton (1992) · Zbl 0838.53053
[13] Herbig, H.-C.: Variations on homological reduction. PhD thesis, Universität Frankfurt a.M. (2007). arXiv:0708.3598
[14] Knutson A.: Some schemes related to the commuting variety. J. Algebraic Geom. 14(2), 283–294 (2005) · Zbl 1074.14044
[15] Malgrange, B.: Ideals of Differentiable Functions. Tata Institute of Fundamental Research Studies in Mathematics, No. 3. Tata Institute of Fundamental Research, Bombay (1967) · Zbl 0177.18001
[16] Matsumura, H.: Commutative Ring Theory, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1989) (translated from the Japanese by M. Reid) · Zbl 0666.13002
[17] Sjamaar R., Lerman E.: Stratified symplectic spaces and reduction. Ann. Math. (2) 134(2), 375–422 (1991) · Zbl 0759.58019 · doi:10.2307/2944350
[18] Waldmann S.: Poisson-Geometrie und Deformationsquantisierung. Eine Einführung. Springer, Berlin (2007) · Zbl 1139.53001
[19] Ziegler G.M.: Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152. Springer, New York (1995) · Zbl 0823.52002
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