Abstract
In this survey, we discuss a series of linearization problems — for Poisson structures, Lie algebroids, and Lie groupoids. The last problem involves a conjecture on the structure of proper groupoids. Attempting to prove this by the method of averaging leads to problems concerning almost actions of compact groups and almost invariant submanifolds for compact group actions. The paper ends with a discussion of possible extensions of the convexity theorems for momentum maps of hamiltonian actions of compact groups.
Research partially supported by NSF Grants DMS-96–25122 and DMS-99–71505 and the Miller Institute for Basic Research in Science.
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References
Almeida, R. and Molino, P.: Suites d’Atiyah et feuilletages transversalement complets, C. R. Acad. Sci. Paris Sér. I Math. 300(1985), 13–15.
Atiyah, M. F.: Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14(1982), 1–15.
Bochner, S.: Compact groups of differentiable transformations, Ann. of Math. 46(1945), 372–381.
Cannas da Silva, A. and Weinstein, A., Geometric models for noncommutative algebras, Berkeley Math. Lecture Notes, Amer. Math. Soc., Providence, 1999.
Conn, J.F.: Normal forms for analytic Poisson structures, Ann. of Math. 119(1984), 576–601.
Conn, J.F.: Normal forms for smooth Poisson structures, Ann. of Math. 121(1985), 565–593.
Dazord, P.: Feuilletages à singularités, Nederl. Akad. Wetensch. Indag. Math.47 (1985), 21–39.
de la Harpe, P. and Karoubi, M.: Représentations approchées d’un groupe dans une algèbre de Banach, Manuscripta Math. 22(1977), 293–310.
Desolneux-Moulis, N.: Linéarisation de certaines structures de Poisson de classe C ∞, Séminaire de géométrie, 1985–1986, Publ. Dép. Math. Nouvelle Sér.B, 86–4, Univ. Claude-Bernard, Lyon, 1986, pp. 55–68.
Flato, M. and Simon, J.: On a linearization program of nonlinear field equations. Phys. Lett. B 94(1980), 518–522.
Grove, K. and Karcher, H.: How to conjugate enclose group actions, Math. Z 132(1973), 11–20.
Grove, K., Karcher, H., and Ruh, E. A.: Group actions and curvature, Invent. Math. 23(1974), 31–48.
Guillemin, V. W. and Sternberg, S.: Remarks on a paper of Hermann, Trans. Amer. Math. Soc. 130(1968), 110–116.
Guillemin, V. and Sternberg, S.: Convexity properties of the moment mapping, Invent. Math. 67(1982), 491–513.
Guillemin, V. and Sternberg, S.: Convexity properties of the moment mapping II, Invent. Math. 77(1984), 533–546.
Hermann, R.: The formal linearization of a semisimple Lie algebra of vector fields about a singular point, Trans. Amer. Math. Soc. 130(1968), 105–109.
Kirwan, F.: Convexity properties of the moment mapping III, Invent. Math. 77(1984), 547–552.
Kushnirenko, A. G.: Linear-equivalent action of a semisimple Lie group in the neighborhood of a fixed point, Fund. Anal. Appl. 1(1967), 89–90.
Mackenzie, K.: Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Notes Series, 124, Cambridge University Press, Cambridge-New York, 1987.
Mikami, K. and Weinstein, A.: Moments and reduction for symplectic groupoids, Publ. Res. Inst. Math. Sci. 24(1988), 121–140.
Ulam, S.: Sets, numbers, and universes: selected works, W. A. Beyer et al., (eds.), The MIT Press, Cambridge, Mass.-London, 1974.
Wade, A.: Normalisation formelle de structures de Poisson, C. R. Acad. Sci. Paris Sér. I Math. 324(1997), 531–536.
Weinstein, A.: Lectures on symplectic manifolds, Regional conference series in mathematics 29, American Mathematical Society, Providence, R.I., 1977.
Weinstein, A.: From Riemann geometry to Poisson geometry and back again, Lecture at Chern Symposium, MSRI, 1998, available at http://msri.org/publications/video/contents.html.
Weinstein, A., Almost invariant submanifolds for compact group actions, J. Eur. Math. Soc. 2(2000), 53–86.
Weinstein, A.: Proper groupoids (in preparation).
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Weinstein, A. (2000). Linearization problems for Lie algebroids and Lie groupoids. In: Dito, G., Sternheimer, D. (eds) Conférence Moshé Flato 1999. Mathematical Physics Studies, vol 21/22. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-1276-3_24
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DOI: https://doi.org/10.1007/978-94-015-1276-3_24
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