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An obstruction to quantizing compact symplectic manifolds
Author(s):
Mark
J.
Gotay;
Janusz
Grabowski;
Hendrik
B.
Grundling
Journal:
Proc. Amer. Math. Soc.
128
(2000),
237-243.
MSC (1991):
Primary 81S99;
Secondary 17B66
Posted:
May 20, 1999
MathSciNet review:
1622742
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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:
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- [Av2]
- Avez, A. [1974-1975] Remarques sur les automorphismes infinitésimaux des variétés symplectiques compactes. Rend. Sem. Mat. Univers. Politecn. Torino, 33, 5-12. MR 53:6628
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- [GG]
- 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.
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 . Trans. Am. Math. Soc. 348, 1579-1597. MR 96h:81036 - [GGT]
- Gotay, M.J., Grundling, H., & Tuynman, G.T. [1996] Obstruction results in quantization theory. J. Nonlinear Sci. 6, 469-498. MR 97j:58054
- [Gr1]
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 and Poisson algebras. Studia Math. 81, 259-270. MR 87c:17021 - [MR]
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Additional Information:
Mark
J.
Gotay
Affiliation:
Department of Mathematics, University of Hawaii, 2565 The Mall, Honolulu, Hawaii 96822
Email:
gotay@math.hawaii.edu
Janusz
Grabowski
Affiliation:
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland
Email:
Janusz.Grabowski@mimuw.edu.pl
Hendrik
B.
Grundling
Affiliation:
Department of Pure Mathematics, University of New South Wales, P.O. Box 1, Kensington, New South Wales, 2033 Australia
Email:
hendrik@maths.unsw.edu.au
DOI:
10.1090/S0002-9939-99-05007-8
PII:
S 0002-9939(99)05007-8
Keywords:
Symplectic manifolds,
quantization,
obstructions
Received by editor(s):
March 11, 1998
Posted:
May 20, 1999
Communicated by:
Peter Li
Copyright of article:
Copyright
1999,
American Mathematical Society
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