nLab
Lie integration

Idea

Lie integration is a process in that assigns to a Lie algebra $𝔤$ – or more generally an ∞-Lie algebra or ∞-Lie algebroid – a Lie group – or more generally ∞-Lie groupoid – that is infinitesimally modeled by $𝔤$.

It turns out that if the ∞-Lie algebroids $𝔞$ involved are incarnated dually in the form of their Chevalley-Eilenberg algebras $\mathrm{CE}(𝔞)$ then the bare ∞-groupoid integrating them (i.e. ignoring the smooth structure) is effectively given by the Sullivan construction applied to the dg-algebra $𝔞$.

This statement was made explicit for dg-Lie algebras in

• Vladimir Hinich, Descent of Deligne groupoids, Internat. Math. Res. Notices, 5:223–239, 1997 (doi, preprint ddg.pdf)

and for general ∞-Lie algebras

• Ezra Getzler, Lie theory for nilpotent $L_{\mathrm{\infty }}$ algebras, (math.AT/0404003)

(whose main point is the discussion of a gauge condition applicable for nilpotent $L_{\mathrm{\infty }}$-algebras that cuts down the result of the Sullivan construction to a much smaller but equivalent model)

and

• Andre Henriques, Integrating $L_{\mathrm{\infty }}$ algebras, Compos. Math. 144 (2008), no. 4, 1017–1045 (doi,math.AT/0603563)

(whose origin possibly preceeds that of Getzler’s article and which considers Banach manifold structure on the resulting ∞-groupoids).

For general ∞-Lie algebroids the general idea has been indicated in

• Pavol Ševera, Some title containing the words “homotopy” and “symplectic”, e.g. this one (arXiv:math.SG/0105080).

There is an evident generalization of the Sullivan construction viewed this way that yields ∞-Lie groupoids (i.e. including the smooth structure). This is discussed at ∞-Lie groupoid.

The traditional Lie integration of Lie algebras and Lie algebroids to Lie groups and Lie groupoids (including the smooth structure) is indeed a special case is exhibited in

• Marius Crainic, Rui Fernandes, Integrability of Lie brackets (arXiv:math.DG/0105033)

Definition

Given an ∞-Lie algebroid $𝔞$ (for instance a Lie algebra, or a Lie algebroid or an L-∞-algebra) and $d\in \mathbb{N}$, a $d$-path in the $\mathrm{\infty }$-Lie algebroid is a morphism of $\mathrm{\infty }$-Lie algebroids

$$\Sigma :T{\Delta }_{\mathrm{Diff}}^d\to 𝔞$$\Sigma : T \Delta^d_{Diff} \to \mathfrak{a}

from the tangent Lie algebroid $T{\Delta }_{\mathrm{Diff}}^d$ of the standard smooth $d$-simplex to $𝔞$.

These $d$-paths naturally form a simplicial set

$$\exp (𝔞):=(\cdots \mathrm{Hom}(T{\Delta }^2,𝔞)\stackrel{t}{\overrightarrow{\to }}\mathrm{Hom}(T{\Delta }^1,𝔞)\overrightarrow{\to }\mathrm{Hom}(T{\Delta }^0,𝔤))$$\exp(\mathfrak{a}) := \left( \cdots Hom(T \Delta^2, \mathfrak{a}) \stackrel{\t}{\stackrel{\to}{\to}} Hom(T \Delta^1, \mathfrak{a}) \stackrel{\to}{\to} Hom(T \Delta^0, \mathfrak{g}) \right)

which is a Kan complex under mild technical fine-tuning of the definition of $d$-paths.

Since morphisms of ∞-Lie algebroids are dually equivalent to dg-algebra morphisms of their Chevalley-Eilenberg algebra, the above is equivalent to

$$\exp (𝔞):=(\cdots \mathrm{Hom}(\mathrm{CE}(𝔞),{\Omega }^{•}({\Delta }^2))\overrightarrow{\overrightarrow{\to }}\mathrm{Hom}(\mathrm{CE}(𝔞),{\Omega }^{•}({\Delta }^1))\overrightarrow{\to }\mathrm{Hom}(\mathrm{CE}(𝔞),{\Omega }^{•}({\Delta }^0))).$$\exp(\mathfrak{a}) := ( \cdots Hom(CE(\mathfrak{a}), \Omega^\bullet(\Delta^2)) \stackrel{\to}{\stackrel{\to}{\to}} Hom(CE(\mathfrak{a}), \Omega^\bullet(\Delta^1)) \stackrel{\to}{\to} Hom(CE(\mathfrak{a}), \Omega^\bullet(\Delta^0)) ) \,.

Here we used that the Chevalley–Eilenberg algebra of the tangent Lie algebroid is the de Rham complex of differential forms. This is recognized as the Sullivan construction in rational homotopy theory for $\mathrm{CE}(mathvraka)$.

This gives the universal $\mathrm{\infty }$-groupoid integrating $𝔞$. If $𝔞$ is $n$-truncated then this construction will not yield in general an $n$-truncated ∞-groupoid $\exp (𝔞)$. Instead one wants to truncate it to

$${\tau }_n\exp (𝔞).$$\tau_n \exp(\mathfrak{a}) \,.

Examples

For more on this see (for the moment) Lie integrated ∞-Lie groupoids.

Integration of Lie algebras

In

• Marius Crainic, Rui Fernandes, Integrability of Lie brackets (arXiv)

it is effectively shown that for $𝔤$ a (finite-dimensional) Lie algebra we have

$${\tau }_1\exp (𝔤)=BG$$\tau_1 \exp(\mathfrak{g}) = \mathbf{B}G

is the delooping one-object groupoid of the simply connected Lie group $G$ corresponding to $G$ under Lie's three theorems.

Integration of Lie 2-algebras

If ${𝔤}_{\mu }$ is the string Lie 2-algebra it is effectively shown in

• André Henriques, Integrating $L_{\mathrm{\infty }}$ algebras,(arXiv)

that

$${\tau }_2\exp ({𝔤}_{\mu })\simeq B\mathrm{String}(G)$$\tau_2 \exp(\mathfrak{g}_\mu) \simeq \mathbf{B}String(G)

is the string 2-group.