∞-Lie theory (higher geometry)
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
There is an obvious functor
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 $Lie$.
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.)
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 $f = Lie(F)$.
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:
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.
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.
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
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
Lie II for Lie groupoids now reads exactly as Lie II for Lie groups, with “simply connected” replaced by “source simply connected”.
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]
Lie II for Lie groupoids was proven in
and
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
On L-infinity algebras related to smooth infinity-groups in higher Lie theory:
(see also references at Lie integration)
Last revised on September 18, 2024 at 01:11:47. See the history of this page for a list of all contributions to it.