nLab
Lie's three theorems

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Lie’s three theorems

There is an obvious functor

Lie:LieGpLieAlg 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 LieLie.

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

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

Remarks

In his third theorem, Lie proved only the existence of 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 LieLie 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 simplLieGroups_{simpl} be the full subcategory of LieGroupsLieGroups consisting of simply connected Lie groups. Then the above implies that restricted to LieGroups simplLieGroups_{simpl}, the functor LieLie 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:LieGroupoidsLieAlgebroids. 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)s^{-1}(x) of the source map s:C 1C 0s : C_1 \to C_0 at every object xC 0x \in C_0 of the Lie groupoid CC) are simply connected. One says

(Cissourcesimplyconnected)(xC 0:π 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 GG integrating a Lie algebroid 𝔤\mathfrak{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).

Revised on May 10, 2013 22:20:03 by Todd Trimble (67.81.93.26)