Lie–Palais theorem

Template:Short description 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. Template:Harvs proved it as a global form of an earlier local theorem due to Sophus Lie.

Statement

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

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

The example of the vector field $d/dx$ 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. Template:Harvtxt gives the following example due to Omori: consider the Lie algebra $𝔤$ of vector fields of the form $f(x,y)\partial /\partial x+g(x,y)\partial /\partial y$ acting on the torus $M={ℝ}^2/{\mathbb{Z}}^2$ such that $g(x,y)=0$ for $0\le x\le 1/2$. This Lie algebra is not the Lie algebra of any group.

Infinite-dimensional generalization

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

References

• Template:Citation Reprinted in collected works volume 5.
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