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On Lie algebra actions

  • * Corresponding author: R. H. Cushman

    * Corresponding author: R. H. Cushman 
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  • In this paper we define an action of a Lie algebra on a smooth manifold. We get nearly the same results as those for group actions, when the flows of the symmetry vector fields are complete. We show that the orbit space of a Lie algebra action is a differential space. We discuss differential spaces occuring in the reduction of symmetries in integrable Hamiltonian systems.

    Mathematics Subject Classification: Primary: 37J15, 37J45; Secondary: 22F05.


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  • [1] N. Aronszajn, Subcartesian and subRiemannian spaces, Notices American Mathematical Society, 14 (1967), 111-111. 
    [2] R. Cushman and J. Śniatycki, Differential structure of orbit spaces, Canad. Math. J., 54 (2001), 715-755.  doi: 10.4153/CJM-2001-029-1.
    [3] R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems, second edition, Birkhäuser, Basel, 2015. doi: 10.1007/978-3-0348-0918-4.
    [4] R. S. Palais, A Global Formulation of the Lie Theory of Transformation Groups, Memoir 22, American Mathematical Society, Providence, R.I. 1957.
    [5] T. Ratiu, C. Wacheux and N. T. Zung, Convexity of singular affine structures and toric-focus integrable Hamiltonian systems, arXiv: 1706.01093v1.
    [6] J. ŚniatyckiDifferential Geometry of Singular Spaces and Reduction of Symmetry, Cambridge University Press, Cambridge, UK, 2013.  doi: 10.1017/CBO9781139136990.
    [7] H. Sussmann, Orbits of families of vector fields and foliations with singularities, Trans. Amer. Math. Soc., 180 (1973), 171-188.  doi: 10.1090/S0002-9947-1973-0321133-2.
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