An obstruction to quantizing compact symplectic manifolds

Gotay, Mark; Grabowski, Janusz; Grundling, Hendrik

doi:10.1090/S0002-9939-99-05007-8

An obstruction to quantizing compact symplectic manifolds
HTML articles powered by AMS MathViewer

by Mark J. Gotay, Janusz Grabowski and Hendrik B. Grundling

Proc. Amer. Math. Soc. 128 (2000), 237-243

DOI: https://doi.org/10.1090/S0002-9939-99-05007-8

Published electronically: May 20, 1999

PDF | Request permission

Abstract:

We prove that there are no nontrivial finite-dimensional Lie representations of certain Poisson algebras of polynomials on a compact symplectic manifold. This result is used to establish the existence of a universal obstruction to quantizing a compact symplectic manifold, regardless of the dimensionality of the representation.

References

André Avez, Représentation de l’algèbre de Lie des symplectomorphismes par des opérateurs bornés, C. R. Acad. Sci. Paris Sér. A 279 (1974), 785–787 (French). MR 362402
André Avez, Remarques sur les automorphismes infinitésimaux des variétés symplectiques compactes, Rend. Sem. Mat. Univ. e Politec. Torino 33 (1974/75), 5–12 (1976). MR 0402814
Filippini, R.J. [1995] The symplectic geometry of the theorems of Borel-Weil and Peter-Weyl. Thesis, University of California at Berkeley.
Ginzburg, V.L. & Montgomery, R. [1997] Geometric quantization and no-go theorems. Preprint dg-ga/9703010.
Mark J. Gotay, On a full quantization of the torus, Quantization, coherent states, and complex structures (Białowieża, 1994) Plenum, New York, 1995, pp. 55–62. MR 1403146
Gotay, M.J. & Grundling, H. [1997] Nonexistence of finite-dimensional quantizations of a noncompact symplectic manifold. To appear in: Differential Geometry and Its Applications 1998, Slovák, J. and Kowalski, O., Eds. (Masaryk University, Brno). Preprint dg-ga/9710024.
Mark J. Gotay, Hendrik Grundling, and C. A. Hurst, A Groenewold-van Hove theorem for $S^2$, Trans. Amer. Math. Soc. 348 (1996), no. 4, 1579–1597. MR 1340175, DOI 10.1090/S0002-9947-96-01559-0
M. J. Gotay, H. B. Grundling, and G. M. Tuynman, Obstruction results in quantization theory, J. Nonlinear Sci. 6 (1996), no. 5, 469–498. MR 1411344, DOI 10.1007/s003329900019
J. Grabowski, Isomorphisms and ideals of the Lie algebras of vector fields, Invent. Math. 50 (1978/79), no. 1, 13–33. MR 516602, DOI 10.1007/BF01406466
Janusz Grabowski, The Lie structure of $C^\ast$ and Poisson algebras, Studia Math. 81 (1985), no. 3, 259–270. MR 808568, DOI 10.4064/sm-81-3-259-270
Jerrold E. Marsden and Tudor S. Ratiu, Introduction to mechanics and symmetry, Texts in Applied Mathematics, vol. 17, Springer-Verlag, New York, 1994. A basic exposition of classical mechanical systems. MR 1304682, DOI 10.1007/978-1-4612-2682-6
Arkadi L. Onishchik, Topology of transitive transformation groups, Johann Ambrosius Barth Verlag GmbH, Leipzig, 1994. MR 1266842