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Lie's three theorems

Lie’s three theorems

There is an obvious functor

$$\mathrm{diff}:\mathrm{Lie}\mathrm{Gp}\to \mathrm{Lie}\mathrm{Alg}$$diff : Lie Gp \to Lie Alg

which sends every Lie group to its Lie algebra and every homomorphism of Lie groups to the corresponding homomorphism of Lie algebras.

Lie’s three theorems establish the following properties of this functor.

1. Lie’s first theorem is today regarded as lacking a good notion of differentiable manifold;

2. Lie II If $G$ and $H$ are Lie groups with Lie algebras $𝔤=\mathrm{diff}(G)$ and $𝔥=\mathrm{diff}(H)$; such that $G$ is simply connected; and if $f:𝔤\to 𝔥$ is a morphism of Lie algebras; then there is a unique morphism $F:G\to H$ of Lie groups lifting $g$, i.e. such that $f=\mathrm{diff}(F)$.

3. Lie III $\mathrm{diff}$ is surjective on objects: to every Lie algebra $𝔤$ there is a Lie group $G$ such that $𝔤=\mathrm{diff}(G)$. Moreover, there exists such $G$ which is simply connected.

Andrew Stacey: Is $\mathrm{diff}$ a standard name for this functor? It seems a little prone to being confused with the diffeomorphism group of $G$ as a smooth manifold.

Toby: I would call the latter functor $\mathrm{Diff}$, but I agree that it's rather similar. The problem is that the standard name for $\mathrm{diff}$, as far as I can tell, is \mathfrak{\lowercase{$-$}}, if you see what I mean, and that's an even less convenient name.

Restriction to simply connected Lie groups

Let ${\mathrm{LieGroups}}_{\mathrm{simpl}}$ be the full subcategory of $\mathrm{LieGroups}$ on simply connected Lie groups. Then the above implies that restricted to ${\mathrm{LieGroups}}_{\mathrm{simpl}}$ the functor $\mathrm{diff}$ becomes a surjective equivalence of categories

Generalization of Lie’s theorems to Lie groupoids

The horizontal categorification of Lie’s theorems for Lie groups leads to analogous statements for Lie groupoids: in this case $\mathrm{diff}$ becomes the differentiation functor from Lie groupoids to Lie algebroids

$$\mathrm{diff}:\mathrm{LieGroupoids}\to \mathrm{LieAlgebroids}.$$diff : LieGroupoids \to LieAlgebroids \,.

In the case of Lie groupoids, the condition of a group being simply connected which plays an important role in the above statements is generalized to the condition that source fibers of the Lie groupoid (the preimages $s^{-1}(x)$ of the source map $s:C_1\to C_0$ at every object $x\in C_0$ of the Lie groupoid $C$) are simply connected. One says

$$(C\mathrm{is}\mathrm{source}-\mathrm{simply}\mathrm{connected})\iff (\forall x\in C_0:{\pi }_1(s^{-1}(x))=0).$$(C is source-simply connected) \Leftrightarrow (\forall x \in C_0 : \pi_1(s^{-1}(x)) = 0 ) \,.

Lie II for Lie groupoids now reads exactly as Lie II for Lie groups, with “simply connected” replaced by “source simply connected”.

Lie II for Lie groupoids was proven in

• K. C. H. Mackenzie and P. Xu, Integration of Lie bialgebroids, Topology, 39(3):445-467

and

• I. Moerdijk and J Mrčun, On integrability of infinitesimal actions, Amer. J. Math. 124(3):567-593, 2002

Lie III for Lie groupoids does not hold in direct generalization:

by the general mechanism of Lie integration the space of morphisms of the source simply-connected topological groupoid $G$ integrating a Lie algebroid $𝔤$ is a quotient space. This quotient may fail to be a manifold due to singularities.

The precise conditions under which Lie algebroids do have Lie groupoids integrating them were found in

• Crainic and Fernandes, Integrability of Lie brackets (arXiv).

A comprehensive review is in

• Rui Loja Fernandes, Marius Crainic, Lectures on Integrability of Lie Brackets, (arxiv)

A review of Lie theory of Lie groupoids in on pages 3-5 of

• Marius Crainic, Rui Loja Fernandes Lectures on Integrability of Lie Brackets (arxiv)

and in the introduction of

• Chenchang Zhu, Lie II theorem for Lie algebroids via stacky Lie groupoids, (arXiv).

Motivation for generalized smooth groupoids

This failure of Lie III for Lie groupoids, i.e. for internal groupoids in Diff seems to suggest that the category of manifolds is not the natural home for general Lie theory. More concretely, it seems to suggest that Lie theory ought to be practiced internal to some category of generalized smooth spaces.

One such choice is given by replacing manifolds by differentiable stacks.

Generalization of Lie’s theorems to stacky Lie groupoids

The generalization of Lie’s theorems from Lie groups to to stacky Lie groupoids is discussed in

• Chenchang Zhu, Lie II theorem for Lie algebroids via stacky Lie groupoids, (arXiv).