nLab Lie's three theorems

Lie’s three theorems

There is an obvious functor

$$Lie \;\colon\; Lie Gp \to Lie Alg$$

from LieGroups to LieAlgebras – Lie differentiation – which sends any Lie group to its Lie algebra and every group homomorphism of Lie groups to the corresponding algebra 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 Élie Cartan and others extended the theorems to give information on actual (“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:

• Hans Duistermaat, Johan A. C. Kolk, Chapter 1 of: Lie groups, Springer (2000) [doi:10.1007/978-3-642-56936-4]

Remark

In his third theorem, Lie proved only the existence of a local Lie group, but not the global existence (nor the simply connected choice) which were established only a few decades later by Élie Cartan. Hence the full theorem would properly be 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 the lectures of Serre 1964.

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”.

Related concepts

• Lie’s third theorem for Leibniz algebras

References

For Lie groups and Lie algebras

Review:

• Jean-Pierre Serre: Lie Algebras and Lie Groups – 1964 Lectures given at Harvard University, Lecture Notes in Mathematics 1500 (1992) [doi:10.1007/978-3-540-70634-2]

• Sigurdur Helgason, Thm. 7.5 in: Differential geometry, Lie groups and symmetric spaces, Graduate Studies in Mathematics 34 (2001) [ams:gsm-34]

For Lie groupoids and Lie algebroids

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:

• Marius Crainic, Rui Loja Fernandes, Integrability of Lie brackets, Ann. of Math. 157 2 (2003) 575-620 [arXiv:math.DG/0105033, doi:10.4007/annals.2003.157.575]

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

• Hsian-Hua Tseng, Chenchang Zhu, Integrating Lie algebroids via stacks, Compositio Mathematica 142 (2006) 251–270 [doi:10.1112/S0010437X05001752]

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:

• Rui Loja Fernandes, Marius Crainic: Lectures on Integrability of Lie Brackets, Geometry & Topology Monographs 17 (2011) 1–107 [arxiv:math.DG/0611259, doi:10.2140/gtm.2011.17.1]

and in the introduction of

• Chenchang Zhu, Lie II theorem for Lie algebroids via stacky Lie groupoids [arXiv:math/0701024]

In higher Lie theory

On L-infinity algebras related to smooth infinity-groups in higher Lie theory:

(see also references at Lie integration)

• Christopher L. Rogers, Jesse Wolfson: Lie’s Third Theorem for Lie $\infty$-Algebras [arXiv:2409.08957]