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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Group actions on spin manifolds

Author: G. Chichilnisky
Journal: Trans. Amer. Math. Soc. 172 (1972), 307-315
MSC: Primary 53C50; Secondary 57D15, 83.53
MathSciNet review: 0309033
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Abstract: A generalization of the theorem of V. Bargmann concerning unitary and ray representations is obtained and is applied to the general problem of lifting group actions associated to the extension of structure of a bundle. In particular this is applied to the Poincaré group $ \mathcal{P}$ of a Lorentz manifold $ M$. It is shown that the topological restrictions needed to lift an action in $ \mathcal{P}$ are more stringent than for actions in the proper Poincaré group $ \mathcal{P}_ \uparrow ^ + $. Similar results hold for the Euclidean group of a Riemannian manifold.

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  • [1] V. Bargmann, On unitary ray representations of continuous groups, Ann. of Math. (2) 59 (1954), 1–46. MR 0058601
  • [2] K. Bichteler, Global existence of spin structures for gravitational fields, J. Mathematical Phys. 9 (1968), 813-815.
  • [3] P. Chernoff and J. Marsden, Hamiltonian systems and quantum mechanics (in preparation).
  • [4] G. Chichilnisky, Group actions on spin manifolds, Thesis, Berkeley, 1970.
  • [5] A. Crumeyrolle, Structures spinorielles, Ann. Inst. H. Poincaré Sect. A (N.S.) 11 (1969), 19–55 (French). MR 0271856
  • [6] David G. Ebin, The manifold of Riemannian metrics, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 11–40. MR 0267604
  • [7] Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
  • [8] Robert Geroch, Spinor structure of space-times in general relativity. I, J. Mathematical Phys. 9 (1968), 1739–1744. MR 0234703
  • [9] Marvin J. Greenberg, Lectures on algebraic topology, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0215295
  • [10] André Haefliger, Sur l’extension du groupe structural d’une espace fibré, C. R. Acad. Sci. Paris 243 (1956), 558–560 (French). MR 0084775
  • [11] Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962. MR 0143793
  • [12] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR 0152974
  • [13] J. E. Marsden, Hamiltonian systems with spin, Canad. Math. Bull. 12 (1969), 203–208. MR 0247848
  • [14] J. Milnor, The representation rings of some classical groups, Mimeographed notes, Princeton University, Princeton, N. J., 1963.
  • [15] -, Notes on characteristic classes, Mimeographed notes, Princeton University, Princeton, N. J., 1957.
  • [16] J. Milnor, Spin structures on manifolds, Enseignement Math. (2) 9 (1963), 198–203. MR 0157388
  • [17] S. B. Myers and N. E. Steenrod, The group of isometries of a Riemannian manifold, Ann. of Math. (2) 40 (1939), no. 2, 400–416. MR 1503467, 10.2307/1968928
  • [18] Richard S. Palais, Seminar on the Atiyah-Singer index theorem, With contributions by M. F. Atiyah, A. Borel, E. E. Floyd, R. T. Seeley, W. Shih and R. Solovay. Annals of Mathematics Studies, No. 57, Princeton University Press, Princeton, N.J., 1965. MR 0198494
  • [19] D. J. Simms, Lie groups and quantum mechanics, Lecture Notes in Mathematics, No. 52, Springer-Verlag, Berlin-New York, 1968. MR 0232579
  • [20] Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
  • [21] Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258
  • [22] E. Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. of Math. (2) 40 (1939), no. 1, 149–204. MR 1503456, 10.2307/1968551
  • [23] E. P. Wigner, Unitary representations of the inhomogeneous Lorentz group including reflections, Group theoretical concepts and methods in elementary particle physics (Lectures Istanbul Summer School Theoret. Phys., 1962) Gordon and Breach, New York, 1964, pp. 37–80. MR 0170976

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Keywords: Bargmann theorem, extension of structure, spinors, lifted action, projective representation, Poincaré group, Euclidean group
Article copyright: © Copyright 1972 American Mathematical Society