Lie–Palais theorem
In differential geometry, the Lie–Palais theorem states that an action of a finite-dimensional Lie algebra on a smooth compact manifold can be lifted to an action of a finite-dimensional Lie group. For manifolds with boundary the action must preserve the boundary; in other words, the vector fields on the boundary must be tangent to the boundary. Palais (1957) proved it as a global form of an earlier local theorem due to Sophus Lie.
The example of the vector field d/dx on the open unit interval shows that the result is false for non-compact manifolds.
Without the assumption that the Lie algebra is finite-dimensional the result can be false. (Milnor 1984) gives the following example due to Omori: the Lie algebra is all vector fields f(x, y) ∂/∂x + g(x, y) ∂/∂y acting on the torus R2/Z2 such that g(x, y) = 0 for 0 ≤ x ≤ 1/2. This Lie algebra is not the Lie algebra of any group. (Pestov 1995) gives an infinite-dimensional generalization of the Lie–Palais theorem for Banach–Lie algebras with finite-dimensional center.
References
- Milnor, John Willard (1984), "Remarks on infinite-dimensional Lie groups", Relativity, groups and topology, II (Les Houches, 1983), Amsterdam: North-Holland, pp. 1007–1057 Reprinted in collected works volume 5.
- Palais, Richard S. (1957), "A global formulation of the Lie theory of transformation groups", Memoirs of the American Mathematical Society 22: iii+123, ISBN 978-0-8218-1222-8, ISSN 0065-9266
- Pestov, Vladimir (1995), "Regular Lie groups and a theorem of Lie-Palais", Journal of Lie Theory 5 (2): 173–178, ISSN 0949-5932, Bibcode: 1994funct.an..3004P
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