×

The prequantization representations of the Poisson Lie algebra. (English) Zbl 0527.58021


MSC:

53D50 Geometric quantization
53C80 Applications of global differential geometry to the sciences
58A12 de Rham theory in global analysis
17B65 Infinite-dimensional Lie (super)algebras
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Avez, A., Cas de réductivitié des algèbres de Lie des symplectomorphismes, C. R. Acad. Sci. Paris Ser. A, 274, 1729-1732 (1972) · Zbl 0243.58002
[2] Avez, A.; Lichnerowicz, A., Dérivations et premier groupe de cohomologie pour les algèbres de Lie attachées à une variété symplectique, C. R. Acad. Sci. Paris Ser. A, 275, 113-118 (1972) · Zbl 0243.58001
[3] Avez, A.; Lichnerowicz, A.; Diaz Miranda, A., Sur l’algèbra des automorphismes infinitésimaux d’une variété symplectique, J. Differential Geom., 9, 1-40 (1974) · Zbl 0283.53033
[4] Blattner, R., Lectures at the Summer School on Nonlinear Partial Differential Operators and Quantization Procedures (July 1981), Institute Für Theoretische Physik der Technischen Universität Clausthal
[5] Dirac, P. A.M, The fundamental equations of quantum mechanics, (Proc. Roy. Soc. London Ser. A, 109 (1926)), 642-653 · JFM 51.0729.01
[6] DuMortier, F.; Takens, F., Characterization of compactness for symplectic manifolds, Bol. Soc. Brasil Mat., 4, 167-173 (1973) · Zbl 0378.58004
[7] Flato, M.; Lichnerowicz, A.; Sternheimer, D., Déformations 1-différentiables des algèbras de Lie attachées a une variété symplectique ou de contact, Composito Math., 31, 47-82 (1975) · Zbl 0317.53039
[8] Hermann, R., (Vector Bundles in Mathematical Physics, Vol. II (1970), Benjamin: Benjamin New York) · Zbl 0213.23603
[9] van Hove, L., Sur certaines représentations unitaires d’un groupe infini de transformations, Acad. Roy. Belg. Cl. Sci. Mem. Collect. 8°, 29, 1-102 (1951) · Zbl 0045.38701
[10] Koopman, B. O., Hamiltonian systems and transformations in Hilbert space, (Proc. Nat. Acad. Sci. U.S.A., 17 (1931)), 315-318 · Zbl 0002.05701
[11] Kostant, B., Quantization and unitary representations, (Tamm, C. T., Lectures in Modern Analysis and Applications, III. Lectures in Modern Analysis and Applications, III, Lecture Notes in Math. No. 170 (1979), Springer-Verlag: Springer-Verlag Berlin/New York), 87-208
[12] Lichnerowicz, A., Cohomologie 1-différentiable des algèbres de Lie attachées a une variété symplectique ou de contact, J. Math Pures Appl., 53, 459-484 (1974) · Zbl 0317.53038
[13] von Neumann, J., Zur Operatorenmethode in der Klassichen Mechanik, Ann. of Math., 33, 587-642 (1932) · Zbl 0005.12203
[14] Palais, R., A global formulation of the Lie theory of transformation groups, Mem. Amer. Math. Soc., 22 (1975) · Zbl 0178.26502
[15] Palais, R., The cohomology of Lie rings, (Proc. Symp. Pure Math., III (1961)), 130-137 · Zbl 0126.03404
[16] Palais, R., Logarithmically exact differential forms, (Proc. Amer. Math. Soc., 12 (1961)), 50-52 · Zbl 0196.38903
[17] Palais, R., (Seminar on the Atiyah-Singer index theorem. Seminar on the Atiyah-Singer index theorem, Ann. of Math. Studies, No. 57 (1965), Princeton Univ. Press: Princeton Univ. Press Princeton, N. J.) · Zbl 0137.17002
[18] Segal, I. E., Quantization of non-linear systems, J. Math. Phys., L, 468-488 (1960) · Zbl 0099.22402
[19] Souriau, J. M., Quantification geometrique, Comm. Math. Phys., 1, 374-398 (1966) · Zbl 1148.81307
[20] Streater, R. F., Canonical quantization, Comm. Math. Phys., 2, 354-374 (1966) · Zbl 0178.28402
[21] Urwin, R., Geometric Quantization and the Cohomology of Lie Algebras, (Thesis (December 1979), Univ. of California at Los Angeles) · Zbl 0527.58021
[22] Warner, F., Foundations of Differentiable Manifolds and Lie Groups (1971), Scott, Foresman: Scott, Foresman Glenview, Ill. · Zbl 0241.58001
[23] Weil, A., Sur les théorèmes de de Rham, Comment. Math. Helv., 26, 119-145 (1952) · Zbl 0047.16702
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.