# nLab Lie's three theorems

Contents

### Context

#### $\infty$-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# Contents

## Lie’s three theorems

There is an obvious functor

$Lie : Lie Gp \to Lie Alg$

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 can be understood as establishing salient properties of this functor. More exactly, Lie’s theorems provide a foundation establishing an equivalence between local Lie groups and Lie algebras; subsequent work by Elie Cartan and others extended the theorems to give information on (global) Lie groups via the functor $Lie$.

1. Lie’s first theorem is purely local; see the Encyclopedia of Math for a statement. (Here one lacks a good notion of differentiable manifold for extending this to a global result.)

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

3. Lie III (Cartan-Lie theorem) The functor $Lie$ is essentially surjective on objects: for every finite dimensional real Lie algebra $\mathfrak{g}$ there is a real Lie group $G$ such that $\mathfrak{g} \cong Lie(G)$. Moreover, there exists such $G$ which is simply connected.

For a classical account see:

###### Remarks

In his third theorem, Lie proved only the existence of a local Lie group, but not the global existence (nor simply connected choice) which were established a few decades later by Elie Cartan. Hence the full theorem is properly called the Cartan-Lie theorem. From an nPOV, the third Lie theorem establishes the essential surjectivity of the functor $Lie$ from the category of local Lie groups to the category of finite dimensional real Lie algebras, and similarly the second Lie theorem establishes that this functor is fully faithful (so the two together establish that this functor is an equivalence). The historically incorrect naming of the Cartan-Lie theorem as the “third Lie theorem” is largely due to the influence of a book based on lectures of Jean-Pierre Serre (Lie algebras and Lie groups, W.A. Benjamin, 1965).

## Restriction to simply connected Lie groups

Let $LieGroups_{simpl}$ be the full subcategory of $LieGroups$ consisting of simply connected Lie groups. Then the above implies that restricted to $LieGroups_{simpl}$, the functor $Lie$ becomes an 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 other words, there are analogous properties for the differentiation functor

$diff : LieGroupoids \to LieAlgebroids.$

from Lie groupoids to Lie algebroids.

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

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

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 $\mathfrak{g}$ is a quotient space. This quotient may fail to be a manifold due to singularities.

On the precise conditions under which Lie algebroids do have Lie groupoids integrating them:

Enlarging Lie groupoids to groupoids in the category of etale stacks and smooth maps results in a Cartan–Lie theorem for Lie algebroids:

In particular, a Lie algebroid can be integrated to an ordinary Lie groupoid if and only if the integrating groupoid in etale stacks is representable.

Comprehensive review:

and in the introduction of

Last revised on March 20, 2023 at 13:59:11. See the history of this page for a list of all contributions to it.