Lie–Palais theorem

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.

Statement

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

If ${\displaystyle M}$ is a manifold with boundary, the statement holds true if the action ${\displaystyle a}$ 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 ${\displaystyle d/dx}$ 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. Milnor (1984, p. 1048) gives the following example due to Omori: consider the Lie algebra ${\displaystyle {\mathfrak {g}}}$ of vector fields of the form ${\displaystyle f(x,y)\partial /\partial x+g(x,y)\partial /\partial y}$ acting on the torus ${\displaystyle M=\mathbb {R} ^{2}/\mathbb {Z} ^{2}}$ such that ${\displaystyle g(x,y)=0}$ for ${\displaystyle 0\leq x\leq 1/2}$. This Lie algebra is not the Lie algebra of any group.

Infinite-dimensional generalization

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, MR 0830252 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, MR 0121424
• Pestov, Vladimir (1995), "Regular Lie groups and a theorem of Lie-Palais", Journal of Lie Theory, 5 (2): 173–178, arXiv:funct-an/9403004, Bibcode:1994funct.an..3004P, ISSN 0949-5932, MR 1389427