Lie–Palais theorem explained

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. proved it as a global form of an earlier local theorem due to Sophus Lie.

Statement

Let

ak{g} be a finite-dimensional Lie algebra and M a closed manifold, i.e. a compact smooth manifold without boundary. Then any infinitesimal action a: ak{g} \to ak{X}(M) of ak{g} on M can be integrated to a smooth action of a finite-dimensional Lie group G , i.e. there is a smooth action \Phi: G x M \to M such that a (\alpha) = d _{e} \Phi ( ⋅ ,x) (\alpha) for every \alpha \in ak{g} .

If

M is a manifold with boundary, the statement holds true if the action a preserves the boundary; in other words, the vector fields on the boundary must be tangent to the boundary.

Counterexamples

d/dx 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. gives the following example due to Omori: consider the Lie algebra

ak{g} of vector fields of the form f(x,y) \partial/\partial x + g(x,y) \partial/\partial y acting on the torus M = R ^{2/Z} ^{2} such that g (x,y) = 0 for 0 \leq x \leq 1 / 2 . This Lie algebra is not the Lie algebra of any group.

Infinite-dimensional generalization

gives an infinite-dimensional generalization of the Lie–Palais theorem for Banach–Lie algebras with finite-dimensional center.

References

• Reprinted in collected works volume 5.