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: |
[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. Śniatycki, Differential 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.![]() ![]() ![]() |