Abstract
Starting from the work by F. A. Berezin, and earlier paper by the author defined an invariant star product on every nonexceptional Kähler symmetric space. In this Letter a recursion formula is obtained to calculate the corresponding invariant Hochschild 2-cochains for spaces of types II and III. An invariant star product is defined on every integral symplectic (Kähler) homogeneous space of simply-connected compact Lie groups (on every integral orbit of the coadjoint representation). The invariant 2-cochains are obtained from the Bochner-Calabi function of the space. The leading term of the lth-2-cochain is determined by the l-power of the Laplace operator.
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References
Flato, M., Lichnerowicz, A., and Sternheimer, D., Comp. Math. 31, 47–82 (1975).
Flato, M., Lichnerowicz, A., and Sternheimer, D., J. Math. Phys. 17, 1754–1762 (1976).
Vey, J., Commun. Math. Helv. 50, 421–454 (1975).
Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D., Ann. Phys. 111, 61–110, 111–151 (1978).
Karasev, M. V. and Maslov, V. P., Russian Math. Surveys 39, 135–205 (1985).
Fronsdal, C., Rep. Math. Phys. 15, 111–145 (1978).
Moreno, C., Lett. Math. Phys. 11, 361–372 (1986).
Helmstetter, J., Cahiers Mathématiques de Montpellier Nos. 25 and 34.
Berezin, F. A., Izv. Akad. Nauk. SSSR. Ser. Math. 38, No. 5 (1974).
Berezin, F. A., Izv. Akad. Nauk. SSSR Ser. Math. 39, No. 2 (1975).
Moreno, C. and Ortega, P., Lett. Math. Phys. 7, 181–193 (1983) and Ann. Inst. Henri. Paincaré 38, 215–241 (1983).
Moreno, C., ‘Invariant Star Products on Kähler Symmetric Spaces; Preprint, Séminaire Lichnerowicz, au Collège de France, 18/2/1986.
Onofri, E., J. Math. Phys. 16, 1087 (1975).
Cahen, M. and Gutt, S., ‘Regular Star Representations of Compact Lie Groups’, Preprint, 1982.
Cahen, M. and Gutt, S., ‘An Algebraic Construction of Star Products on the Regular Orbits of Semisimple Lie Groups’, in a book in honor of Ibor Robinson, to appear 1986.
Sternheimer, D., ‘Phase-space Representations’, Lecture at the 14th AMS-SIAM Summer Sem., Chicago, 1982, Preprint.
A. Crumeyrolle, C.R. Acad. Sci. Paris 282, A-1367–1976; Actes des Journées Relativistes de Grenoble, 15–17 May 1981.
Bayen, F., Lecture Notes in Phys. 94, p. 266, Springer-Verlag, Berlin, 1978.
Hua, L. K., Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Amer. Math. Soc., 1963.
Combet, E., Lectures Notes in Math. 937, Springer-Verlag, Berlin, 1982.
Fedoryuk, M. V., Russian Math. Surveys 26, 1 (1971).
Lichnerowicz, A., Actes Coll. Int. de Géométrie Différentielle, Strasbourg (1953), pp. 171–184.
Borel, A., Proc. Nat. Acad. Sci. U.S.A. 40, 1147–1151 (1954).
Serre, J. P., Séminaire Bourbaki, No. 100 (1954), Paris.
Takeuchi, M., Japan J. Math. 4, 171–219 (1978).
Calabi, E., Ann. Math. 58, 1–23 (1953).
Perelomov, A. M., Commun. Math. Phys. 26, 222–236 (1972).
Rawnsley, J. H., Quart. J. Math. Oxford (2), 28, 403–415 (1977).
Arnal, D., Pacific J. Math. 114, 285 (1984).
Arnal, D. and Cortet, J. C., Lett. Math. Phys. 9, 25 (1985).
Lugo, V. A., Lett. Math. Phys. 5, 506 (1981).
Gutt, S., Thèse, Université Libre de Bruxelles, 1979–1980.
Cahen, M. and Gutt, S., Lett. Math. Phys. 5, 219 (1981).
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Moreno, C. Invariant star products and representations of compact semisimple Lie groups. Lett Math Phys 12, 217–229 (1986). https://doi.org/10.1007/BF00416512
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DOI: https://doi.org/10.1007/BF00416512