Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Openness and convexity for momentum maps


Authors: Petre Birtea, Juan-Pablo Ortega and Tudor S. Ratiu
Journal: Trans. Amer. Math. Soc. 361 (2009), 603-630
MSC (2000): Primary 53D20
DOI: https://doi.org/10.1090/S0002-9947-08-04689-8
Published electronically: September 9, 2008
MathSciNet review: 2452817
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this paper is finding the essential attributes underlying the convexity theorems for momentum maps. It is shown that they are of a topological nature; more specifically, we show that convexity follows if the map is open onto its image and has the so-called local convexity data property. These conditions are satisfied in all the classical convexity theorems and hence they can, in principle, be obtained as corollaries of a more general theorem that has only these two hypotheses. We also prove a generalization of the so-called Local-to-Global Principle that only requires the map to be closed and to have a normal topological space as domain, instead of using a properness condition. This allows us to generalize the Flaschka-Ratiu convexity theorem to noncompact manifolds.


References [Enhancements On Off] (What's this?)

  • 1. R. ABRAHAM, J. E. MARSDEN, AND T. S. RATIU, Manifolds, Tensor Analysis, and Applications, second enlarged edition, Applied Mathematical Sciences 75, Springer Verlag, 1989. MR 960687 (89f:58001)
  • 2. A. ALEKSEEV, On Poisson actions of compact Lie groups on symplectic manifolds, Journ. Diff. Geom., 45(1997), pp. 241-256. MR 1449971 (99b:58086)
  • 3. M. F. ATIYAH, Convexity and commuting Hamiltonians, Bull. London Math. Soc., 14(1982), pp. 1-15. MR 642416 (83e:53037)
  • 4. A. M. BLOCH, H. FLASCHKA, AND T. S. RATIU, Schur-Horn-Kostant convexity theorem for the diffeomorphism group of the annulus, Invent. Math. 113 (1993), no. 3, pp. 511-529. MR 1231835 (94i:58063)
  • 5. G. E. BREDON, Introduction to Compact Transformation Groups, Academic Press, New York and London, 1972. MR 0413144 (54:1265)
  • 6. C. O. CHRISTENSON AND W. L. VOXMAN, Aspects of Topology, Marcel Dekker, 1977. MR 0487938 (58:7521)
  • 7. M. P. CONDEVAUX, P. DAZORD AND P. MOLINO, Géométrie du moment, Travaux du Séminaire Sud-Rhodanien de Géométrie, I, Publ. Dép. Math. Nouvelle Sér. B, 88-1, Univ. Claude-Bernard, Lyon, 1988, 131-160. MR 1040871
  • 8. J. J. DUISTERMAAT AND J. A. KOLK, Lie Groups. Universitext, Springer-Verlag, 1999. MR 2265844 (2007j:22016)
  • 9. R. ENGELKING, General Topology. Second edition. Sigma Series in Pure Mathematics, 6. Heldermann Verlag, Berlin, 1989. MR 1039321 (91c:54001)
  • 10. H. FLASCHKA AND T. S. RATIU, A convexity theorem for Poisson actions of compact Lie groups, Ann. Sci. Ecole Norm. Sup., 4ème série, 29(6) (1995), pp. 787-809. MR 1422991 (98a:58068)
  • 11. V. GUILLEMIN AND S. STERNBERG, Convexity properties of the moment mapping, Invent. Math., 67(1982), pp. 491-513. MR 664117 (83m:58037)
  • 12. V. GUILLEMIN AND S. STERNBERG, Convexity properties of the moment mapping II, Invent. Math., 77(1984), pp. 533-546. MR 759258 (86b:58042a)
  • 13. V. GUILLEMIN AND S. STERNBERG, A normal form for the moment map, Differential Geometric Methods in Mathematical Physics, S. Sternberg ed., Mathematical Physics Studies, 6. Reidel Publishing Co., 1984, pp. 161-175. MR 767835 (86d:58033)
  • 14. J. HILGERT, K.-H. NEEB, AND W. PLANK, Symplectic convexity theorems and coadjoint orbits, Compositio Math. 94(1994), pp. 129-180. MR 1302314 (96d:53053)
  • 15. J. G. HOCKING AND G. S. YOUNG, Topology, Addison-Wesley Publishing Company, Inc., 1961. MR 0125557 (23:A2857)
  • 16. A. HORN, Eigenvalues of sums of Hermitian matrices, Pacific J. Math., 12(1962), pp. 225-241. MR 0140521 (25:3941)
  • 17. W. Y. HSIANG, Lectures on Lie Groups. Series on University Mathematics, World Scientific, Vol. 2, 2000. MR 1788014 (2002d:22001)
  • 18. Y. KARSHON AND E. LERMAN, The centralizer of invariant functions and divison properties of the moment map, Illinois J. Math. 41(3) (1997), pp. 462-487. MR 1458185 (98e:58077)
  • 19. G. KIRCHHOFF, Vorlesungen ueber Mathematische Physik, Vol I, Kapitel 20, Teubner, Leipzig, 1883.
  • 20. F. KIRWAN, Convexity properties of the moment mapping III, Invent. Math., 77(1984), pp. 547-552. MR 759257 (86b:58042b)
  • 21. V. L. KLEE, JR., Convex sets in linear spaces, Duke Math. J., 18(1951), pp. 443-466. MR 0044014 (13:354f)
  • 22. A. W. KNAPP, Lie Groups Beyond an Introduction. Progress in Mathematics, volume 140. Birkhäuser, 1996. MR 1399083 (98b:22002)
  • 23. F. KNOP, Convexity of Hamiltonian manifolds, J. Lie Theory, 12(2002), pp. 571-582. MR 1923787 (2003j:53131)
  • 24. A. KNUTSON AND T. TAO, Honeycombs and sums of Hermitian matrices, Notices of the AMS, 48(2) (2001), pp. 175-186. MR 1811121 (2002g:15020)
  • 25. B. KOSTANT, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Ecole Norm. Sup., 4ème série, 6(1973), pp. 413-455. MR 0364552 (51:806)
  • 26. E. LERMAN, Symplectic cuts, Mathematical Research Letters, 2(1995), pp. 247-258. MR 1338784 (96f:58062)
  • 27. E. LERMAN, E. MEINRENKEN, S. TOLMAN, AND C. WOODWARD, Nonabelian convexity by symplectic cuts, Topology 37(1998), pp. 245-259. MR 1489203 (99a:58069)
  • 28. C.-M. MARLE, Modèle d'action hamiltonienne d'un groupe de Lie sur une variété symplectique, Rend. Sem. Mat. Univ. Politecn. Torino, 43(2)(1995), pp. 227-251. MR 859857 (88a:58075)
  • 29. S. MAZUR, Über konvexe Mengen in linearen normierten Räumen, Studia Math., 4(1933), 70-84.
  • 30. J. MONTALDI AND T. TOKIEDA, Openness of momentum maps and persistence of extremal relative equilibria, Topology, 42(2003), pp. 833-844. MR 1958531 (2004b:37109)
  • 31. A. NEUMANN, An infinite-dimensional version of the Schur-Horn convexity theorem, J. Funct. Anal. 161 (1999), no. 2, pp. 418-451. MR 1674643 (2000a:22030)
  • 32. J.-P. ORTEGA AND T. S. RATIU, Momentum Maps and Hamiltonian Reduction, Progress in Mathematics 222, Birkhäuser Boston, 2004. MR 2021152 (2005a:53144)
  • 33. E. PRATO, Convexity properties of the moment map for certain non-compact manifolds, Comm. Anal. Geom., 2(2) (1994), pp. 267-278. MR 1312689 (95k:58065)
  • 34. I. SCHUR, Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie, Sitzungsber. der Berliner Math. Gesellschaft, 22(1923), pp. 9-20.
  • 35. R. SJAMAAR, Convexity properties of the moment map re-examined, Adv. Math., 138(1) (1998), 46-91. MR 1645052 (2000a:53148)
  • 36. A. WEINSTEIN, Poisson geometry of discrete series orbits, and momentum convexity for noncompact group actions, Letters in Mathematical Physics, 56(1) (2001), 17-30. MR 1848163 (2002m:53130)
  • 37. G. T. WHYBURN, Topological Analysis, Princeton University Press, 1964. MR 0165476 (29:2758)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53D20

Retrieve articles in all journals with MSC (2000): 53D20


Additional Information

Petre Birtea
Affiliation: Departamentul de Matematică, Universitatea de Vest, RO–1900 Timişoara, Romania
Email: birtea@math.uvt.ro

Juan-Pablo Ortega
Affiliation: Département de Mathématiques de Besançon, Université de Franche-Comté, UFR des Sciences et Techniques, 16 route de Gray, F–25030 Besançon cédex, France
Email: Juan-Pablo.Ortega@math.univ-fcomte.fr

Tudor S. Ratiu
Affiliation: Centre Bernoulli, École Polytechnique Fédérale de Lausanne, CH–1015 Lausanne, Switzerland
Email: tudor.ratiu@epfl.ch

DOI: https://doi.org/10.1090/S0002-9947-08-04689-8
Received by editor(s): July 11, 2006
Published electronically: September 9, 2008
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society