In differential geometry, a field of mathematics, the Lie–Palais theorem is a partial converse to the fact that any smooth action of a Lie group induces an infinitesimal action of its Lie algebra.
Palais (1957) proved it as a global form of an earlier local theorem due to Sophus Lie.
be a finite-dimensional Lie algebra and
a closed manifold, i.e. a compact smooth manifold without boundary.
Then any infinitesimal action
can be integrated to a smooth action of a finite-dimensional Lie group
is a manifold with boundary, the statement holds true if the action
preserves the boundary; in other words, the vector fields on the boundary must be tangent to the boundary.
The example of the vector field
on the open unit interval shows that the result is false for non-compact manifolds.
Similarly, without the assumption that the Lie algebra is finite-dimensional, the result can be false.
Milnor (1984, p. 1048) gives the following example due to Omori: consider the Lie algebra
of vector fields of the form
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.