# American Institute of Mathematical Sciences

April  2020, 13(4): 1115-1129. doi: 10.3934/dcdss.2020066

## On Lie algebra actions

 Department of Mathematics and Statistics, University of Calgary, Calgary, AB, T2N 1N4, Canada

* Corresponding author: R. H. Cushman

Received  October 2017 Revised  July 2018 Published  April 2019

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.

Citation: Richard H. Cushman, Jędrzej Śniatycki. On Lie algebra actions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1115-1129. doi: 10.3934/dcdss.2020066
##### References:
 [1] N. Aronszajn, Subcartesian and subRiemannian spaces, Notices American Mathematical Society, 14 (1967), 111-111.   Google Scholar [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.  Google Scholar [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.  Google Scholar [4] R. S. Palais, A Global Formulation of the Lie Theory of Transformation Groups, Memoir 22, American Mathematical Society, Providence, R.I. 1957.  Google Scholar [5] T. Ratiu, C. Wacheux and N. T. Zung, Convexity of singular affine structures and toric-focus integrable Hamiltonian systems, arXiv: 1706.01093v1. Google Scholar [6] J. Śniatycki, Differential Geometry of Singular Spaces and Reduction of Symmetry, Cambridge University Press, Cambridge, UK, 2013.  doi: 10.1017/CBO9781139136990.  Google Scholar [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.  Google Scholar

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##### References:
 [1] N. Aronszajn, Subcartesian and subRiemannian spaces, Notices American Mathematical Society, 14 (1967), 111-111.   Google Scholar [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.  Google Scholar [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.  Google Scholar [4] R. S. Palais, A Global Formulation of the Lie Theory of Transformation Groups, Memoir 22, American Mathematical Society, Providence, R.I. 1957.  Google Scholar [5] T. Ratiu, C. Wacheux and N. T. Zung, Convexity of singular affine structures and toric-focus integrable Hamiltonian systems, arXiv: 1706.01093v1. Google Scholar [6] J. Śniatycki, Differential Geometry of Singular Spaces and Reduction of Symmetry, Cambridge University Press, Cambridge, UK, 2013.  doi: 10.1017/CBO9781139136990.  Google Scholar [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.  Google Scholar
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