Lie integration

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###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
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#### Integration theory
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[[!include integration theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

_Lie integration_ is a process that assigns to a [[Lie algebra]] $\mathfrak{g}$ -- or more generally to an [[L-∞-algebra|∞-Lie algebra]] or [[∞-Lie algebroid]] -- a [[Lie group]] -- or more generally [[∞-Lie groupoid]] -- that is [[infinitesimal space|infinitesimally]] modeled by $\mathfrak{g}$. It is essentially the reverse operation to [[Lie differentiation]], except that there are in general several objects Lie integrating a given Lie algebraic datum, due to the fact that the infinitesimal data does not uniquely determine global topological properties.

Classically, Lie integration of [[Lie algebras]] is part of [[Lie's three theorems]], which in particular finds an unique (up to [[isomorphism]]) [[simply connected]] Lie group integrating a given finite-dimensional Lie algebra.

One may observe that the [[simply connected]] [[Lie group]] integrating a (finite-dimensional) Lie algebra is equivalently realized as the collection of [[equivalence classes]] of [[Lie algebra valued 1-forms]] on the interval where two such are identified if they are interpolated by a _flat_ Lie-algebra valued 1-form on the [[disk]]. ([Duistermaat-Kolk 00, section 1.14](#DuistermaatKolk00), see also the example [below](#LieAlgebrasToLieGroups)).

This _path method_ of Lie integration stands out as having natural generalizations to [[higher Lie theory]] ([Ševera 01](#Severa01)).

In its evident generalization from [[Lie algebra valued differential forms]] to [[Lie algebroid valued differential forms]] this provides a means for Lie integration of [[Lie algebroids]] (e.g. [Crainic-Fernandes 01](#Crainic)).

In another direction, one may observe that [[L-∞ algebras]] are [[formal dual|formally dually]] incarnated by their [[Chevalley-Eilenberg algebra|Chevalley-Eilenberg]] [[dg-algebras]], and that under this identification the evident generalization of the path method to [[L-∞ algebra valued differential forms]] is essentially the [[Sullivan construction]], known from [[rational homotopy theory]], applied to these dg-algebras ([Hinich 97](#Hinich97), [Getzler 04](#Getzler04)). Or rather, the bare such construction gives the [[geometrically discrete ∞-groupoid|geometrically discrete]] [[∞-group]] underlying what should be the Lie integration to a [[smooth ∞-group]]. This is naturally obtained, as in the classical case, by suitably smoothly parameterizing the [[∞-Lie algebroid valued differential forms]] ([Henriques 08](#Henriques), [Roytenberg 09](#Roytenberg09), [FSS 12](#FSS12)).

Both these directions may be combined via the evident concept of [[∞-Lie algebroid valued differential forms]] to yield a Lie integration of [[∞-Lie algebroids]] to [[smooth ∞-groupoids]]. (Moreover, the same formula directly generalizes from $L_\infty$-algebroids to [[A-infinity categories]] to yield the _[[dg-nerve]]_ construction.)

While the construction exists and behaves as expected in examples, there is to date no good general theory providing higher analogs of, say, [[Lie's three theorems]]. But people are working on it.

## Definition

Throughout, let $\mathfrak{a}$ be an [[∞-Lie algebroid]] (for instance a [[Lie algebra]], or a [[Lie algebroid]] or an [[L-∞-algebra]]). Write

$$
  CE(\mathfrak{a}) \in dgAlg
$$

for its [[Chevalley-Eilenberg algebra]], a [[dg-algebra]]. Notice that, by the discussion at _[[L-∞ algebra]]_ and at _[[∞-Lie algebroid]]_, the Chevalley-Eilenberg dg-algebras $CE(\mathfrak{a})$ is the [[formal dual]] of $\mathfrak{a}$, in that the [[functor]]

$$
  CE \colon \infty LieAlgd \hookrightarrow dgAlg^{op}
$$

is a [[fully faithful functor]]. Indeed, the following definition of Lie integration (being just a smooth refinement of the [[Sullivan construction]]) makes sense just as well for any dg-algebra, not necessarily in the [[essential image]] of this embedding. But only for dg-algebras in the essential image of this embedding do the examples come out as expected for [[higher Lie theory]].

An [[∞-Lie algebra]] is equivalently a [[pointed object|pointed]] [[∞-Lie algebroid]] whose base space is the point. We write $\mathfrak{b}\mathfrak{g} \in \infty Lie Algd$ for objects of this form ("[[delooping]]")

$$
  \array{
     \infty LieAlgd &\stackrel{CE}{\hookrightarrow}& dgAlg^{op}
     \\
     \uparrow^{\mathrlap{b}} & \nearrow_{\mathrlap{CE}}
     \\
     \infty LieAlg
     \,.
  }
$$

Notice that this induces some degree shifts that may be a little ambiguous in situations like the [[line Lie n-algebra]]: as an [[L-∞ algebra]] this is $b^{n-1}\mathbb{R}$, the corresponding [[∞-Lie algebroid]] is $b^n \mathbb{R}$.

### Higher dimensional paths in an $\infty$-Lie algebroid
 {#HigherPathsInAnInfinityLieAlgebroid}

+-- {: .num_example}
###### Example

For $X$ a [[smooth manifold]] (possibly [[nLab:manifold with boundary|with boundary]] and [[manifold with corners|with corners]]) then its [[tangent Lie algebroid]] $T X$ is the one whose [[Chevalley-Eilenberg algebra]] is the [[de Rham complex]]

$$
  CE(T X) = (\Omega^\bullet(X), d_{dR})
  \,.
$$

=--

+-- {: .num_defn #kPath}
###### Definition

For $k \in \mathbb{N}$ write $\Delta^k$ for the standard $k$-[[simplex]] regarded as a [[smooth manifold]] ([[nLab:manifold with boundary|with boundary]] and [[manifold with corners|with corners]]).

A **$k$-path** in the $\infty$-Lie algebroid $\mathfrak{a}$ is a morphism of $\infty$-Lie algebroids of the form

$$
  \Sigma \; \coloneqq \; T \Delta^k \longrightarrow \mathfrak{a}
$$

from the [[tangent Lie algebroid]] $T \Delta^k$ of the standard smooth $k$-simplex to $\mathfrak{a}$. Dually this is equivalently a [[homomorphism]] of [[dg-algebras]]

$$
  \Omega^\bullet(\Delta^k) \longleftarrow CE(\mathfrak{a}) \;\colon\; \Sigma^*
$$

from the [[Chevalley-Eilenberg algebra]] of $\mathfrak{a}$ to the [[de Rham complex]] of $\Delta^d$.

=--

See also at _[[differential forms on simplices]]_.

+-- {: .num_remark}
###### Remark

A $k$-path in $\mathfrak{a}$, def. \ref{kPath}, is equivalently

1. a _flat_ $\mathfrak{a}$-[[infinity-Lie algebroid valued differential form|valued differential form]] on $\Delta^k$;

1. a [[Maurer-Cartan element]] in $\Omega^\bullet(\Delta^k)\otimes \mathfrak{a}$.

=--

The Lie integration of $\mathfrak{a}$ is essentially the [[simplicial object]] whose $k$-cells are the $d$-paths in $\mathfrak{a}$. However, in order for this to be well-behaved, it is possible and useful to restrict to $d$-paths that are sufficiently well-behaved towards the [[boundary]] of the simplex:

+-- {: .num_defn}
###### Definition

Regard the smooth simplex $\Delta^k$ as embedded into the [[Cartesian space]] $\mathbb{R}^{k+1}$ in the standard way, and equip $\Delta^k$ with the [[metric space]] structure induced this way.

A smooth [[differential form]] $\omega$ on $\Delta^k$ is said to have **sitting instants** along the boundary if, for every $(r \lt k)$-face $F$ of $\Delta^k$ there is an [[open neighbourhood]] $U_F$ of $F$ in $\Delta^k$ such that $\omega$ restricted to $U$ is constant in the directions perpendicular to the $r$-face on its value restricted to that face.

More generally, for any $U \in $ [[CartSp]] a smooth differential form $\omega$ on $U \times\Delta^k$ is said to have sitting instants if there is $0 \lt \epsilon \in \mathbb{R}$ such that for all points $u : * \to U$ the pullback along  $(u, \mathrm{Id}) : \Delta^k \to U \times \Delta^k$ is a form with sitting instants on $\epsilon$-[[open neighbourhood|neighbourhood]]s of faces.

Smooth forms with sitting instants clearly form a sub-dg-algebra of all smooth forms. We write $\Omega^\bullet_{si}(U \times \Delta^k)$ for this sub-dg-algebra.

We write $\Omega_{si,vert}^\bullet(U \times \Delta^k)$ for the further sub-dg-algebra of [[vertical differential form]]s with respect to the projection $p : U \times \Delta^k \to U$, hence the [[coequalizer]]

$$
  \Omega^\bullet(U)
  \stackrel{\stackrel{p^*}{\longrightarrow}}{\underset{0}{\longrightarrow}}
  \Omega^\bullet_{si}(U \times \Delta^k)
  \to
  \Omega^\bullet_{si, vert}(U \times \Delta^k)
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

The dimension of the normal direction to a face depends on the dimension of the face:  there is one perpendicular direction to a codimension-1 face, and $k$ perpendicular directions to a
vertex.

=--

+-- {: .num_example}
###### Examples

* A smooth 0-form (a [[smooth function]]) has sitting instants on $\Delta^1$ if in a neighbourhood of the endpoints it is constant.

  A smooth function $f : U \times \Delta^1 \to \mathbb{R}$ is in $\Omega^0_{\mathrm{vert}}(U \times \Delta^1)$ if there is $0 \lt \epsilon \in \mathbb{R}$ such that for each $u \in U$ the function $f(u,-) : \Delta^1  \simeq [0,1] \to \mathbb{R}$ is constant on $[0,\epsilon) \coprod (1-\epsilon,1)$.

* A smooth 1-form has sitting instants on $\Delta^1$ if in a neighbourhood of the endpoints it vanishes.

* Let $X$ be a [[smooth manifold]], $\omega \in \Omega^\bullet(X)$ be a smooth differential form. Let

  $$
    \phi \colon \Delta^k \to X
  $$

  be a [[smooth function]] that has [[sitting instants]] as a function: towards any $j$-face of $\Delta^k$ it eventually becomes perpendicularly constant.

  Then the [[pullback of differential forms|pullback]] form $\phi^* \omega \in \Omega^\bullet(\Delta^k)$ is a form with sitting instants.

=--

+-- {: .num_remark}
###### Remark

The condition of sitting instants serves to make smooth differential forms not be affected by the boundaries and corners of $\Delta^k$. Notably for $\omega_j \in \Omega^\bullet(\Delta^{k-1})$ a collection of forms with sitting instants on the $(k-1)$-cells of a [[horn]] $\Lambda^k_i$ that coincide on adjacent boundaries, and for

$$
  p \colon \Delta^k \to \Lambda^{k-1}_i
$$

a standard piecewise smooth [[retract]], the [[pullback of differential forms|pullbacks]]

$$
  p^* \omega_i
$$

glue to a single smooth differential form (with sitting instants) on $\Delta^k$.

=--

+-- {: .num_remark}
###### Remark

That $\omega \in \Omega^\bullet(\Delta^k)$ having sitting instants does not imply that there is a neighbourhood of the boundary of $\Delta^k$ on which $\omega$ is entirely constant. It is important for the following constructions that in the vicinity of the boundary $\omega$ is allowed to vary parallel to the boundary, just not perpendicular to it.

=--

### Integration to a discrete $\infty$-groupoid
 {#IntToBareGrpd}

Here we discuss the [[discrete ∞-groupoid]]s underlying the
[[smooth ∞-groupoid]]s to which an [[∞-Lie algebroid]] integrates.

For $\mathfrak{a}$ an $\infty$-Lie algebroid, the $d$-paths in $\mathfrak{a}$ naturally form a [[simplicial set]] as $d$ varies:

$$
  \begin{aligned}
    \exp(\mathfrak{a})_{bare}
    & \coloneqq
    \left(
     \cdots
     Hom_{\infty LieAlgd}(T \Delta^2, \mathfrak{a})
     \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
     Hom_{\infty LieAlgd}(T \Delta^1, \mathfrak{a})
     \stackrel{\longrightarrow}{\longrightarrow}
     Hom_{\infty LieAlgd}(T \Delta^0, \mathfrak{g})
    \right)
     \\
    & =
   (
   \cdots
   Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(\Delta^2))
   \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
   Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(\Delta^1))
   \stackrel{\longrightarrow}{\longrightarrow}
   Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(\Delta^0))
   )
  \end{aligned}
  \,.
$$

(We are indicating only the face maps, not the degeneracy maps, just for notational simplicity).

If here instead of smooth differential forms one uses [[polynomial differential forms]] then this is precisely the [[Sullivan construction]] of [[rational homotopy theory]] applied to $CE(\mathfrak{a})$. We next realize [[smooth structure]]  on this and hence realize this as an object in [[higher Lie theory]].

+-- {: .num_remark #SpuriousHomotopyGroups}
###### Remark
**(spurious homotopy groups)**

For $\mathfrak{a}$ a [[infinity-Lie algebroid|Lie n-algebroid]] (an $n$-[[truncated]] $\infty$-Lie algebroid) this construction will not yield in general an $n$-[[truncated]] [[∞-groupoid]] $\exp(\mathfrak{a})$.

To see this, consider the example (discussed in detail [below](#LieAlgebrasToLieGroups)) that $\mathfrak{a} = \mathfrak{g}$ is an ordinary [[Lie algebra]]. Then $\exp(\mathfrak{g})_n$ is canonically identified with the set of smooth based maps $\Delta^n \to G$ into the simply connected [[Lie group]] that integrates $\mathfrak{g}$ in ordinary [[Lie theory]]. This means that the simplicial [[homotopy group]]s of $\exp(\mathfrak{g})$ are the topological homotopy groups of $G$, which in general (say for $G$ the [[orthogonal group]] or [[unitary group]]) will be non-trivial in arbitrarily higher degree, even though $\mathfrak{g}$ is just a Lie 1-algebra. This phenomenon is well familiar from [[rational homotopy theory]], where a classical theorem asserts that the rational homotopy groups of $\exp(\mathfrak{g})$ are generated from the generators in a [[minimal Sullivan model]] resolution of $\mathfrak{g}$.

=--

For the purposes of [[higher Lie theory]] therefore instead one wants to [[truncated|truncate]] $\exp(\mathfrak{g})$ to its $(n+1)$-[[coskeleton]]

$$
  \mathbf{cosk}_{n+1}\exp(\mathfrak{a})_{bare}
  \,.
$$

This divides out [[n-morphisms]] by $(n+1)$-morphisms and forgets all higher higher nontrivial morphisms, hence all higher homotopy groups.

$\,$

### Integration to a smooth $\infty$-groupoid
  {#SmoothIntegration}

We now discuss Lie integration of $\infty$-Lie algebroids to [[smooth ∞-groupoid]]s, [[presentable (∞,1)-category|presented]] by the [[model structure on simplicial presheaves]] $[CartSp_{smooth}^{op}, sSet]_{proj,loc}$ over the [[site]] [[CartSp]]${}_{smooth}$.

For the following definition recall the [[presentable (∞,1)-category|presentation]] of [[smooth ∞-groupoids]] by the [[model structure on simplicial presheaves]] over the [[site]] [[CartSp]]${}_{smooth}$.

+-- {: .num_defn}
###### Definition

For $\mathfrak{a}$ an [[L-∞ algebra]] of [[finite type]] with [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ define the [[simplicial presheaf]]

$$
  \exp(\mathfrak{a}) \;\colon\; CartSp_{smooth}^{op} \to sSet
$$

by

$$
  \exp(\mathfrak{a})
  \;\colon\;
  (U,[k]) \mapsto
  Hom_{dgAlg}(CE(\mathfrak{a}),
  \Omega^\bullet(U \times \Delta^k)_{si,vert})
  \,,
$$

for all $U \in $ [[CartSp]] and $[k] \in \Delta$.

=--

+-- {: .num_remark}
###### Remark

Compared to the integration to [[discrete ∞-groupoids]] [above](#IntToBareGrpd) this definition knows about $U$-parametrized _smooth families_ of $k$-paths in $\mathfrak{g}$.

The underlying [[discrete ∞-groupoid]] is recovered as that of the $\mathbb{R}^0 = *$-parameterized family:

$$
  \exp(\mathfrak{a}) \colon \mathbb{R}^0 \mapsto \exp(\mathfrak{a})_{disc}
  \,.
$$

=--

+-- {: .num_prop}
###### Proposition

The objects $\exp(\mathfrak{g})$ are indeed [[Kan complexes]] over each $U \in $ [[CartSp]].

=--

+-- {: .proof}
###### Proof

Observe that the standard [[continuous function|continuous]] [[horn]] [[retracts]] $f : \Delta^k \to \Lambda^k_i$ are [[smooth function|smooth]] away from the preimages of the $(r \lt k)$-faces of $\Lambda[k]^i$.

For $\omega \in \Omega^\bullet_{si,vert}(U \times \Lambda[k]^i)$ a differential form with sitting instants on $\epsilon$-neighbourhoods, let therefore $K \subset \partial \Delta^k$ be the set of points of distance $\leq \epsilon$ from any subface. Then we have a smooth function

$$
  f : \Delta^k \setminus K \to \Lambda^k_i \setminus K
  \,.
$$

The [[pullback of differential forms|pullback]] $f^* \omega \in \Omega^\bullet(\Delta^k \setminus K)$ may be extended constantly back to a form with sitting instants on all of $\Delta^k$.

The resulting assignment

$$
  (CE(\mathfrak{g}) \stackrel{A}{\longrightarrow} \Omega^\bullet_{si,vert}(U \times \Lambda^k_i))
  \mapsto
  (CE(\mathfrak{g}) \stackrel{A}{\to} \Omega^\bullet_{si,vert}(U \times \Lambda^k_i) \stackrel{f^*}{\to} \Omega^\bullet_{si,vert}(U \times \Delta^n))
$$

provides fillers for all [[horns]] over all $U \in $ [[CartSp]].

=--

+-- {: .num_defn}
###### Definition

Write $\mathbf{cosk}_{n+1} \exp(a)$ for the simplicial presheaf obtained by postcomposing $\exp(\mathfrak{a}) : CartSp^{op} \to sSet$ with the $(n+1)$-[[coskeleton]] [[functor]] $\mathbf{cosk}_{n+1} : sSet \stackrel{tr_n}{\longrightarrow} sSet_{\leq n+1} \stackrel{cosk_{n+1}}{\to} sSet$.

=--

## Properties

### Homotopy groups
 {#PropertiesHomotopyGroups}

...([Henriques, theorem 6.4](#Henriques))... remark \ref{SpuriousHomotopyGroups}...

### Quillen adjunction
 {#QuillenAdjunction}

The [above](#SmoothIntegration) construction of Lie integraton to [[smooth ∞-groupoids]] clearly applies to all [[differential graded-commutative algebras]], not necessarily just those which are [[Chevalley-Eilenberg algebras]] of [[L-∞ algebras]]. (but up to [[weak equivalence]], there is no difference). With this generalization, the higher Lie integration extends to a [[Quillen adjunction]] (Prop. \ref{LieIntegrationIsRightQuillenFunctor} below). In order to state this conveniently, we first make more explicit the [[functor]] assigning smooth families of [[smooth differential forms on simplices]] (Def. \ref{SmoothDifferentialFormsOnSimplicesWithSittingInstants} below).

$\,$

+-- {: .num_defn #SmoothDifferentialFormsOnSimplicesWithSittingInstants}
###### Definition
**(smooth families of [[smooth differential forms on simplices]] with [[sitting instants]])**

For $k \in \mathbb{N}$, write $\Delta^k_{mfd}$ for the [[k-simplex]] canonically regarded as a [[smooth manifold with boundaries and corners]].

For $n \in \mathbb{N}$, regard $\mathbb{R}^n \times \Delta^k_{mfd} \overset{p_1}{to} \mathbb{R}^n$ as the [[trivial bundle|trivial]] [[fiber bundle]] over the [[Cartesian space]] $\mathbb{R}^n$ with [[fiber]] that  smooth [[k-simplex]].

Write

$$
  \Omega^\bullet_{vert}(\mathbb{R}^n \times \Delta^k_{mfd})
  \hookrightarrow
  \Omega^\bullet(\mathbb{R}^n \times \Delta^k_{mfd})
$$

for the [[subobject|sub-]][[dgc-algebra]] of the [[de Rham algebra]] on the [[vertical differential forms]] with respect to this bundle structure.

Moreover, write

$$
  \Omega^\bullet_{vert, si}\left(\mathbb{R}^n \times \Delta^k_{mfd}\right)
  \hookrightarrow
  \Omega^\bullet_{vert}\left(\mathbb{R}^n \times \Delta^k_{mfd}\right)
$$

for the further [[subobject|sub-]][[dgc-algebra]] on those [[vertical differential forms]] which have [[sitting instants]] towards the [[boundary]] of the [[k-simplex]].

Via [[pullback of differential forms]] this construction provides a [[functor]]

$$
  \Omega^\bullet_{vert,si}
  \;\colon\;
  CartSp \times \Delta
    \longrightarrow
  dgcAlg_{\mathbb{R}, conn}^{op}
$$

from the [[product category]] of the [[category]] [[CartSp]] of [[Cartesian spaces]] and [[smooth functions]] between them, with the [[simplex category]], to the [[opposite category|opposite]] of the [[category]] of connective [[dgc-algebras]] over the [[real numbers]].

=--

([FSS 12, Def. 4.2.1](#FSS12), see [Braunack-Mayer 18, Def. 3.1.3](#BraunackMayer18))

+-- {: .num_prop #LieIntegrationIsRightQuillenFunctor}
###### Proposition
**(Lie integration is [[right Quillen functor]] to [[smooth ∞-groupoids]])**

There is a [[Quillen adjunction]]

$$
  dgcAlg^{op}_{\mathbb{R}, \geq 0, proj}
    \;
    \underoverset
      {\underset{ Spec }{\longrightarrow}}
      {\overset{ \mathcal{O} }{\longleftarrow}}
      {\phantom{A}\phantom{{}_{Qu}}\bot_{Qu}\phantom{A}}
    \;
  [CartSp^{op},sSet_{Qu}]_{proj,loc}
$$

between

1. the [[projective local model structure on simplicial presheaves]] over [[CartSp]], regarded as a [[site]] via the [[good open cover]] [[coverage]] (i.e. [[locally presentable (∞,1)-category|presenting]] [[smooth ∞-groupoids]]);

1. the [[opposite model category|opposite]] [[projective model structure on connective dgc-algebras]] over the [[real numbers]]

given by [[nerve and realization]] with respect to the [[functor]] of [[smooth differential forms on simplices]]
$
  CartSp \times \Delta
    \overset{\Omega^\bullet_{vert,si}}{\longrightarrow}
  dgcAlg_{\mathbb{R}, conn}^{op}
$ from Def. \ref{SmoothDifferentialFormsOnSimplicesWithSittingInstants}:

1. the [[right adjoint]] $Spec$ sends a [[dgc-algebra]] $A \in dgcAlg_{\mathbb{R},\geq 0}$ to the [[simplicial presheaf]] which in degree $k$ is the set of [[dg-algebra]]-[[homomorphism]] form $A$ into the [[dgc-algebras]] of [[smooth differential forms on simplices]] $\Omega^\bullet_{si,vert}(-)$ (Def. \ref{SmoothDifferentialFormsOnSimplicesWithSittingInstants}):

   $$
     Spec(A)
     \;\colon\;
     \mathbb{R}^n \times \Delta[k]
     \;\mapsto\;
     Hom_{dgcAlg_{\mathbb{R}}}
     \left(
       A , \Omega^\bullet_{si, vert}(\mathbb{R}^n \times \Delta^k_{mfd})
     \right)
   $$

1. the [[left adjoint]] $\mathcal{O}$ is the [[Yoneda extension]] of the [[functor]] $\Omega^\bullet_{vert,si} \;\colon\; CartSp \times \Delta \to dgcAlg_{\mathbb{R},conn}^{op}$ assigning [[dgc-algebras]] of [[smooth differential forms on simplices]] from Def. \ref{SmoothDifferentialFormsOnSimplicesWithSittingInstants},

   hence which acts on a [[simplicial presheaf]] $\mathbf{X} \in [CartSp^{op}, sSet] \simeq [\CartSp^{op} \times \Delta^{op}, Set]$, expanded via the [[co-Yoneda lemma]] as a [[coend]] of [[representable presheaves|representables]], as

   $$
     \mathcal{O}
     \;\colon\;
     \mathbf{X}
     \simeq
     \int^{n,k}
     y(\mathbb{R}^n \times \Delta[k]) \times \mathbf{X}(\mathbb{R}^n)_k
     \;\mapsto\;
     \int_{n,k}
     \underset{\mathbf{X}(\mathbb{R}^n)_k}{\prod}
     \Omega^\bullet_{si,vert}\left(\mathbb{R}^n \times \Delta^k_{mfd}\right)
   $$

=--

([Braunack-Mayer 18, theorem 3.1.10](#BraunackMayer18))

## Examples

See also at _[[smooth ∞-groupoid -- structures]]_ the section _[Exponentiated ∞-Lie algebras](smooth+infinity-groupoid+--+structures#StrucLieAlg)_.

### Interating Lie algebras to Lie groups
  {#LieAlgebrasToLieGroups}

Let $\mathfrak{g} \in L_\infty$ be an ordinary (finite dimensional) [[Lie algebra]]. Standard [[Lie theory]] (see [[Lie's three theorems]]) provides a [[simply connected]] [[Lie group]] $G$ integrating $\mathfrak{g}$.

With $G$ regarded as a [[smooth ∞-groupoid|smooth ∞-group]] write $\mathbf{B}G \in $ [[Smooth∞Grpd]] for its [[delooping]]. The standard presentation of this on $[CartSp_{smooth}^{op}, sSet]$ is by the [[simplicial presheaf]]

$$
  \mathbf{B}G_c
  \colon
  U \mapsto N(C^\infty(U,G) \stackrel{\longrightarrow}{\longrightarrow} *)
  \,.
$$

See at _[smooth infinity-groupoid -- structures -- Lie groups](smooth+infinity-groupoid+--+structures#LieGroups)_ for more details.

+-- {: .num_prop #ResultOfIntegrationOfOrdinaryLieAlgebra}
###### Proposition

The operation of [[parallel transport]] $P \exp(\int -) : \Omega^1([0,1], \mathfrak{g}) \to G$ yields a weak equivalence (in $[CartSp^{op}, sSet]_{proj}$)

$$
  P \exp(\int (-) )
  \;\colon\;
  \mathbf{cosk}_3 \exp(\mathfrak{g})
  \;\simeq\;
   \mathbf{cosk}_2 \exp(\mathfrak{g}) \simeq \mathbf{B}G_c
  \,.
$$

=--

This follows from the [[Steenrod-Wockel approximation theorem]] and the following observation.

+-- {: .num_lemma}
###### Lemma

For $X$ a [[simply connected]] [[smooth manifold]] and $x_0 \in X$ a basepoint, there is a canonical [[bijection]]

$$
  \Omega^1_{flat}(X,\mathfrak{g}) \simeq C^\infty_*(X,G)
$$

between the set of [[Lie-algebra valued 1-form]]s on $X$ whose [[curvature]] 2-form vanishes, and the set of [[smooth function]]s $X\to G$ that take $x_0$ to the neutral element $e \in G$.

=--

+-- {: .proof}
###### Proof

The bijection is given as follows. For $A \in \Omega^1_{flat}(X,\mathfrak{g})$ a flat 1-form, the corresponding function $f_A : X \to G$ sends $x \in X$ to the [[parallel transport]] along any path $x_0 \to x$ from the base point to $x$

$$
  f_A : x \mapsto tra_A(x_0 \to x)
  \,.
$$

Because of the assumption that the [[curvature]] 2-form of $A$ vanishes and the assumption that $X$ is [[simply connected]], this assignment is independent of the choice of path.

Conversely, for every such function $f : X \to G$ we recover $A$ as the pullback of the [[Maurer-Cartan form]] on $G$

$$
  A = f^* \theta
  \,.
$$

=--

From this we obtain

+-- {: .proof}
###### Proof of prop. \ref{ResultOfIntegrationOfOrdinaryLieAlgebra}.

The $\infty$-groupoid $\mathbf{cosk}_2 \exp(\mathfrak{g})$ is equivalent to the [[groupoid]] with a single object (no non-trivial 1-form on the point) whose morphisms are equivalence classes of smooth based paths $\Delta^1 \to G$ (with sitting instants), where two of these are taken to be equivalent if there is a smooth [[homotopy]] $D^2 \to G$ (with sitting instant) between them.

Since $G$ is [[simply connected]], these equivalence classes are labeled by the endpoints of these paths, hence are canonically identified with $G$.

=--

+-- {: .num_remark}
###### Remark

We do not need to fall back to classical [[Lie theory]] to obtain $G$ in the above argument. A detailed discussion of how to find $G$ with its group structure and smooth structure from $d$-paths in $\mathfrak{g}$ is in ([Crainic](#Crainic)).

=--

### Integrating to line/circle Lie $n$-groups
 {#IntegrationToLineNGroup}

+-- {: .num_defn}
###### Definition

For $n \in \mathbb{N}, n \geq 1$ write $b^{n-1} \mathbb{R}$ for the [[L-∞-algebra]] whose [[Chevalley-Eilenberg algebra]] is given by a single generator in degree $n$ and vanishing differential. We may call this the **[[line Lie n-algebra]]**.

Write $\mathbf{B}^{n} \mathbb{R}$ for the <a href="http://ncatlab.org/nlab/show/smooth+infinity-groupoid#CircleLienGroup">smooth line (n+1)-group</a>.

=--

+-- {: .num_prop}
###### Observation

The [[discrete ∞-groupoid]] underlying $\exp(b^{n-1} \mathbb{R})$ is given by the [[Kan complex]] that in degree $k$ has the set of closed differential $n$-forms (with sitting instants) on the $k$-[[simplex]]

$$
  \exp(b^{n-1} \mathbb{R})_{disc}
  :

  [k] \mapsto \Omega^n_{si,cl}(\Delta^k)
$$

=--

+-- {: .num_prop #LieIntegrationOfLinenLieAlgebra}
###### Proposition

The $\infty$-Lie integration of $b^{n-1} \mathbb{R}$ is the [[circle n-group]] $\mathbf{B}^{n} \mathbb{R}$.

Moreover, with $\mathbf{B}^n \mathbb{R}_{chn} \in [CartSp_{smooth}^{op}, sSet]$ the standard presentation given under the [[Dold-Kan correspondence]] by the chain complex of sheaves concentrated in degree $n$ on $C^\infty(-, \mathbb{R})$ the equivalence is induced by the [[fiber integration]] of differential $n$-forms over the $n$-[[simplex]]:

$$
  \int_{\Delta^\bullet}
   :
  \exp(b^{n-1} \mathbb{R})
   \stackrel{\simeq}{\longrightarrow}
  \mathbf{B}^{n} \mathbb{R}_{chn}
  \,.
$$

=--

+-- {: .proof}
###### Proof

First we observe that the map

$$
  \int_{\Delta^\bullet}
  :
  (\omega \in \Omega^n_{si,vert,cl}(U \times \Delta^k))
  \mapsto
  \int_{\Delta^k} \omega \in C^\infty(U, \mathbb{R})
$$

is a morphism of
[[simplicial presheaves]] $\exp(b^{n-1} \mathbb{R}) \to \mathbf{B}^{n}\mathbb{R}_{chn}$ on [[CartSp]]${}_{smooth}$. Since it goes between presheaves of abelian [[simplicial group]]s by the [[Dold-Kan correspondence]] it is sufficient to check that we have a morphism of [[chain complex]]es of presheaves on the corresponding [[Moore complex|normalized chain complex]]es.

The only nontrivial degree to check is degree $n$. Let $\lambda \in \Omega_{si,vert,cl}^n(\Delta^{n+1})$. The differential of the [[Moore complex|normalized chains complex]] sends this to the signed sum of its restrictions to the $n$-faces of the $(n+1)$-simplex. Followed by the integral over $\Delta^n$ this is the piecewise integral of $\lambda$ over the boundary of the $n$-simplex. Since $\lambda$ has sitting instants, there is $0 \lt \epsilon \in \mathbb{R}$ such that there are no contributions to this integral in an $\epsilon$-neighbourhood of the $(n-1)$-faces. Accordingly the integral is equivalently that over the smooth surface inscribed into the $(n+1)$-simplex, as indicated in the following diagram

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Since $\lambda$ is a closed form on the $n$-simplex, this surface integral vanishes, by the [[Stokes theorem]]. Hence $\int_{\Delta^\bullet}$ is indeed a chain map.

It remains to show that $\int_{\Delta^\bullet} : \exp(b^{n-1} \mathbb{R}) \to \mathbf{B}^{n}\mathbb{R}_{chn}$ is an [[isomorphism]] on all the [[simplicial homotopy]]s group over each $U \in CartSp$. This amounts to the statement that

1. a smooth family of closed $n$-forms with sitting instants on the boundary of $\Delta^{n+1}$ may be extended to a smooth family of closed forms with sitting instants on $\Delta^{n+1}$ precisely if their smooth family of integrals over the boundary vanishes;

1. Any smooth family of closed $n \lt k$-forms with sitting instants on the boundary of $\Delta^{k+1}$ may be extended to a smooth family of closed $n$-forms with sitting instants on $\Delta^{k+1}$.

To demonstrate this, we want to work with forms on the $(k+1)$-[[ball]] instead of the $(k+1)$-[[simplex]]. To achieve this, choose again $0 \lt \epsilon \in \mathbb{R}$ and construct the [[diffeomorphism|diffeomorphic]] image of $S^k \times [1,1-\epsilon]$ inside the $(k+1)$-simplex as indicated in the above diagram: outside an $\epsilon$-neighbourhood of the corners the image is a rectangular $\epsilon$-thickening of the faces of the simplex. Inside the $\epsilon$-neighbourhoods of the corners it bends smoothly. By the [[Steenrod-Wockel approximation theorem]] the diffeomorphism from this $\epsilon$-thickening of the smoothed boundary of the simplex to $S^k \times [1-\epsilon,1]$ extends to a smooth function from the $(k+1)$-simplex to the $(k+1)$-ball.

By choosing $\epsilon$ smaller than each of the sitting instants of the given $n$-form on $\partial \Delta^{k+1}$, we have that this $n$-form vanishes on the $\epsilon$-neighbourhoods of the corners and is hence entirely determined by its restriction to the smoothed simplex, identified with the $(k+1)$-ball.

It is now sufficient to show: a smooth family of smooth $n$-forms $\omega \in \Omega^n_{vert,cl}(U \times S^k)$ extends to a smooth family of closed $n$-forms $\hat \omega \in \Omega^n_{vert,cl}(U \times B^{k+1})$ that is radially constant in a neighbourhood of the boundary for all $n \lt k$ and for $k = n$ precisely if its smooth family of integrals vanishes, $\int_{S^k} \omega = 0 \in C^\infty(U, \mathbb{R})$.

Notice that over the point this is a direct consequence of the [[de Rham theorem]] ([[kernel of integration is the exact differential forms]]): an $n$-form $\omega$ on $S^k$ is exact precisely if $n \lt k$  or if $n = k$ and its integral vanishes. In that case there is an $(n-1)$-form $A$ with $\omega = d A$. Choosing any smoothing function $f : [0,1] \to [0,1]$ (smooth, surjective, non-decreasing and constant in a neighbourhood of the boundary) we obtain an $n$-form $f \wedge A$ on $(0,1] \times S^k$, vertically constant in a neighbourhood of the ends of the interval, equal to $A$ at the top and vanishing at the bottom. Pushed forward along the canonical $(0,1] \times S^k \to D^{k+1}$ this defines a form on the $(k+1)$-ball, that we denote by the same symbol $f \wedge A$. Then the form $\hat \omega := d (f \wedge A)$ solves the problem.

To complete the proof we have to show that this simple argument does extend to smooth families of forms, i.e., that we can choose the $(n-1)$-form $A$ in a way depending smoothly on the the $n$-form $\omega$.

One way of achieving this is using [[Hodge theory]]. Fix a [[Riemannian metric]] on $S^n$, and let $\Delta$ be the corresponding [[Laplace operator]], and $\pi$ the projection on the space of [[harmonic forms]]. Then the
<a href="http://ncatlab.org/nlab/show/Hodge+theory#ResultsOverRiemannianManifold">central result of Hodge theory</a> for [[compact topological space|compact]] [[Riemannian manifold]]s states that the operator $\pi$, seen as an operator from the
[[de Rham complex]] to itself, is a cochain map [[homotopy|homotopic]] to the identity, via an explicit homotopy $P := d^* G$ expressed in terms of the adjoint $d^*$ of the de Rham differential and of the [[Green operator]] $G$ of $\Delta$. Since the $k$-form $\omega$ is exact its projection on [[harmonic form]]s vanishes. Therefore

$$
  \begin{aligned}
    \omega & = (Id-\pi)\omega
    \\
     & = d (P\omega)+P (d\omega)
    \\
    & = d (P\omega).
  \end{aligned}
$$

Hence $A := P\omega$ is a solution of the differential equation $d A=\omega$ depending smoothly on $\omega$.

=--

### Integrating the string Lie 2-algebra to the string Lie 2-group

Let $\mathfrak{string} = \mathfrak{g}_\mu$ be the [[string Lie 2-algebra]].

Then $\mathbf{cosk}_3 \exp(\mathfrak{g}_\mu)$ is equivalent to the [[2-groupoid]] $\mathbf{B}String$

* with a single object;

* whose morphisms are based paths in $G$;

* whose 2-morphisms are equivalence class of pairs $(\Sigma,c)$, where

  * $\Sigma : D^2_*  \to G$ is a smooth based map (where we use a [[homeomorphism]] $D^2 \simeq \Delta^2$ which away from the corners is smooth, so that forms with sitting instants there do not see any non-smoothness, and the basepoint of $D^2_*$ is the 0-vertex of $\Delta^2$)

  * and $c \in U(1)$, and where two such are equivalent if the maps coincides at their boundary and if for any 3-ball $\phi : D^3 \to G$ filling them the labels $c_1, c_2 \in U(1)$ differ by the integral $\int_{D^3} \phi^* \mu(\theta) \;\; mod \;\; \mathbb{Z}$,,

where $\theta$ is the [[Maurer-Cartan form]], $\mu(\theta) = \langle \theta\wedge [\theta \wedge \theta]\rangle $ the 3-form obtained by plugging it into the cocycle.

This is the [[string Lie 2-group]]. It's construction in terms of integration by paths is due to ([Henriques](#Henriques))

### Integrating Lie algebroids to (stacky) Lie groupoids

Unlike finite dimensional Lie algebras, not every Lie algebroid may be integrated to a Lie groupoid. There exists topological obstruction coming from $\pi_2(L)$ of the leaf $L\subset M$ of a Lie algebroid $A\to M$. The integrability criteria is completely classified through the behaviour of certain monodromy groups and that is the achievement of  [Crainic-Fernandes 01](#Crainic). These monodromy groups, providing obstructions of integration, may be seen as the image of a transgression map of the long exact sequence of [[Lie algebroid homotopy groups]] induced by the natural [[Lie algebroid fibration]] $A|_L \to L$. (see [Brahic-Zhu 10](#BZ) ).

Let us now describe the construction of the universal groupoid for a Lie algebroid $A$. This is a major step contained in  [Crainic-Fernandes 01](#Crainic) and was earlier discovered in the setting of Poisson manifolds as the phase space of [[Poisson sigma model]] in [Cattaneo-Felder 00](#CattaneoFelder01).

Given a Lie algebroid $A\to M$, an $A$-path (a Lie algebroid path), is a Lie algebroid morphism from $T\Delta^1 \to A$, that is, it is a path $a$ in $A$ such that $a=\rho(\gamma)$ where $\gamma=\pi(a)$ is the base path of $a$ (see [example](https://ncatlab.org/nlab/show/Lie+groupoid#example) ).

An (end-fixing) $A$-homotopy (a Lie algebroid homotopy) is a Lie algebroid morphism from $T\square \to A$ satisfying certain boundary conditions. Let us be more precise: a vector bundle morphism $T\square \to A$ can be denoted by $a dt + b ds$ (see [example](https://ncatlab.org/nlab/show/Lie+groupoid#example_2) ). Then the boundary condition is that $b(s,0)=0$ and $b(s,1)=0$. The fact that $adt+bds$ is a Lie algebroid morphism is equivalent to the following PDE:

$$\partial_t b- \partial_s a= \nabla_{\rho \alpha} \beta - \nabla_{\rho \beta} \alpha +[\alpha, \beta] $$
where $\nabla$ is a TM connection on $A$ and $\alpha$ and $\beta$ are certain time dependent sections of $A$ extending $a$ and $b$ respectively. Notice that there is also a way writing down the right hand side independent of choice of sections. See Section 1 of [Crainic-Fernandes 01](#Crainic).

Then the universal groupoid associated to $A$ is the space of $A$-paths ($P_a A$) dividing by $A$-homotopies. It is naturally a topological groupoid. But only when the obstruction vanishes, it becomes a Lie groupoid and is the source-simply connected Lie groupoid integrating $A$.

Fortunately $A$-homotopies form finite codimensional foliation $F$, even though the quotient might not be always representible. Thus, $Mon_F(P_aA)$ represents a differentiable stack which becomes a stacky Lie groupoid over $M$. It turns out that there is a one-to-one correspondence between \'etale stacky Lie groupoid and Lie algebroid. This correspondence provides a positive answer to [[Lie's third theorem]] for Lie algebroids.  [Tseng-Zhu 04](#TZ).

## Related concepts

* [[Kan-fibrant simplicial manifold]]

* [[parallel transport]], [[higher parallel transport]]

* [[holonomy]], [[higher holonomy]]

  * [[nonabelian Stokes theorem]]

* [[dg-nerve]]

* [[function algebras on infinity-stacks]]

[[!include infinitesimal and local - table]]

## References

The "path method" of integrating Lie algebras to simply connected Lie groups appears in

* {#DuistermaatKolk00} [[Hans Duistermaat]], J. A. C. Kolk, section 1.14 of _Lie groups_, 2000

The idea of identifying the [[Sullivan construction]] applied to [[Chevalley-Eilenberg algebras]] as Lie integration to
[[discrete ∞-groupoid]]s appears in

* {#Hinich97} [[Vladimir Hinich]], _Descent of Deligne groupoids_, Internat. Math. Res. Notices, 5:223–239, 1997 ([doi](http://dx.doi.org/10.1155/S1073792897000160), [preprint ddg.pdf](http://math.haifa.ac.il/hinich/WEB/mypapers/ddg.pdf))

and for general [[∞-Lie algebra]]s in

* {#Getzler04} [[Ezra Getzler]], _Lie theory for nilpotent $L_\infty$ algebras_, Annals of Mathematics, 170 (2009), 271–301 ([math.AT/0404003](http://arxiv.org/abs/math/0404003))

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

This was refined from integration to bare $\infty$-groupoids to an integration to [[internal ∞-groupoid]]s in [[Banach manifold]]s in

* {#Henriques} [[André Henriques]], _Integrating $L_\infty$ algebras_, Compos. Math. __144__  (2008), no. 4, 1017--1045 ([doi](http://dx.doi.org/10.1112/S0010437X07003405),[math.AT/0603563](http://arxiv.org/abs/math.AT/0603563))

(whose origin possibly preceeds that of Getzler's article).

For general [[∞-Lie algebroids]] the general idea of the integration process by "$d$-paths" had been indicated in

* {#Severa01} [[Pavol Severa]],  _[[Some title containing the words "homotopy" and "symplectic", e.g. this one]]_, based on a talk at _[Poisson 2000](http://www.lpthe.jussieu.fr/~dito/poissongeometry/Poisson2000/index.html)_, [CIRM Marseille Luminy](https://www.cirm-math.fr/), June 2000 ([arXiv:0105080](http://arxiv.org/abs/math/0105080))

* Pavol Ševera, Michal Širaň, _Integration of differential graded manifolds_, [arxiv/1506.04898](http://arxiv.org/abs/1506.04898)

Lie integration of [[dg-modules]] to [[smooth spectrum|smooth]] [[parameterized spectra]] ([[twisted cohomology|twisted]] [[differential cohomology theories]]);

* {#BraunackMayer18} [[Vincent Braunack-Mayer]], section 3.1 of _[[schreiber:thesis Braunack-Mayer|Rational parameterized stable homotopy theory]]_, Zurich 2018 ([pdf](https://ncatlab.org/schreiber/show/thesis+Braunack-Mayer#pdf))

Discussion of Lie integration of [[Lie algebroids]] by the path method is due to

* {#Crainic} [[Marius Crainic]], [[Rui Loja Fernandes]], *Integrability of Lie brackets*, Ann. of Math. **157** 2 (2003) 575-620 [[arXiv:math.DG/0105033](http://arxiv.org/abs/math.DG/0105033), [doi:10.4007/annals.2003.157.575](https://doi.org/10.4007/annals.2003.157.575)]

reviewed in

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

and following [Duistermaat-Kolk (2000 section 1.14)](#DuistermaatKolk00), as well as the discussion of the special case of Lie integration of [[Poisson Lie algebroids]] to [[symplectic groupoids]] in

* {#CattaneoFelder01} [[Alberto Cattaneo]], [[Giovanni Felder]], _Poisson sigma models and symplectic groupoids_, in _Quantization of Singular Symplectic Quotients_, (ed. [[Klaas Landsman]], M. Pflaum, M. Schlichenmeier), Progress in Mathematics 198 (Birkhäuser,
2001), 61–93. ([arXiv:math/0003023](http://arxiv.org/abs/math/0003023))

upgraded to the stacky version by

* {#TZ} Tseng Hsiang-Hua, [[Chenchang Zhu]], _Integrating Lie algebroids via stacks_, Compositio Mathematica, Volume 142 (2006), Issue 01, pp 251-270, [arXiv:math/0405003](http://arxiv.org/abs/math/0405003).

A general proof that equivalent $L_\infty$-algebras integrate to equivalent Lie $\infty$-groupoids is in

* [[Christopher Rogers]], [[Chenchang Zhu]], _On the homotopy theory for Lie ∞-groupoids, with an application to integrating L∞-algebras_ ([arXiv:1609.01394](https://arxiv.org/abs/1609.01394))

Obstruction interpreted as transgression of a Lie algebroid fibration by

* {#BZ} Olivier Brahic, [[Chenchang Zhu]], _Lie algebroid fibrations_, Adv. Math. 226 (2011), no. 4, 3105–3135, [arXiv:1001.4904](http://arxiv.org/abs/1001.4904).

A description of Lie integration with values in [[smooth ∞-groupoids]] regarded as [[simplicial presheaves]] on [[CartSp]] (and further the Lie integration of [[infinity-Lie algebra cohomology|L-infinity cocycles]]) is in

* {#FSS12} [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Cech Cocycles for Differential characteristic Classes]]_, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 ([arXiv:1011.4735](http://arxiv.org/abs/1011.4735))

Essentially the same integration prescription is considered in

* {#Roytenberg09} [[Dmitry Roytenberg]], _Differential graded manifolds and associated stacks: an overview_ ([pdf](https://sites.google.com/site/dmitryroytenberg/unpublished/dg-stacks-overview.pdf))

The Lie integration- of [[Lie infinity-algebroid representation|Lie algebroid representations]] $\mathfrak{a} \to end(V)$ to morphisms of [[∞-categories]] $A \to Ch_\bullet^\circ$ / [[higher parallel transport]] is discussed in

* [[Camilo Arias Abad]], [[Florian Schätz]], _The $A_\infty$ de Rham theorem and integration of representations up to homotopy_ ([arXiv:1011.4693](http://arxiv.org/abs/1011.4693))

Application to the problem of Lie integrating ordinary but infinite-dimensional Lie algebras is in

* [[Christoph Wockel]], [[Chenchang Zhu]], _Integrating central extensions of Lie algebras via Lie 2-groups_ ([arXiv:1204.5583](http://arxiv.org/abs/1204.5583))

A generalization of [[Lie integration]] to conjectural Leibniz groups has been conjectured by [[J-L. Loday]]. A local version via local Lie [[rack]]s has been proposed in

* Simon Covez, _The local integration of Leibniz algebras_, [arXiv:1011.4112](http://arxiv.org/abs/1011.4112); _On the conjectural cohomology for groups_, [arXiv:1202.2269](http://arxiv.org/abs/1202.2269);
_L'intégration locale des algèbres de Leibniz_, Thesis (2010), [pdf](http://tel.archives-ouvertes.fr/docs/00/49/54/69/PDF/THESE_Simon_Covez.pdf)

Integration from Lie algebroids to groupoids is also studied in the dual language and generality of integration of Lie-Reinhart algebras
and commutative Hopf algebroids,

* Alessandro Ardizzoni, Laiachi El Kaoutit, Paolo Saracco, _Towards differentiation and integration between Hopf algebroids and Lie algebroids_, [arXiv:1905.10288](https://arxiv.org/abs/1905.10288)

[[!redirects Lie integrations]]

[[!redirects higher Lie integration]]
[[!redirects higher Lie integrations]]

category: Lie theory


Lie n-groupoid

#Contents#
* table of contents
{:toc}

## Idea

The [[geometric infinity-stacks]] within all [[smooth infinity-groupoids]] are called _Lie $\infty$-groupoids_. These are equivalently those smooth $\infty$-groupoids which have a [[representable functor|presentation]] by [[Kan-fibrant simplicial manifolds]]. See there for more.

## Related $n$Lab entries

* [[Lie differentiation]]

[[!redirects Lie n-groupoids]]

[[!redirects Lie infinity-groupoids]]
[[!redirects Lie-infinity-groupoid]]

[[!redirects infinity-Lie-groupoid]]
[[!redirects infinity-Lie-groupoids]]

[[!redirects infinity-Lie groupoid]]
[[!redirects infinity-Lie groupoids]]

[[!redirects Lie ∞-groupoid]]
[[!redirects Lie ∞-groupoids]]

[[!redirects ∞-Lie groupoid]]
[[!redirects ∞-Lie groupoids]]

[[!redirects ∞LieGrpd]]

[[!redirects Lie-infinity groupoid]]
[[!redirects Lie ∞-groupoids]]


Lie operad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given an ambient additive [[symmetric monoidal category]], the _Lie operad_ is the [[operad]] whose [[algebras over an operad]] are [[Lie algebra]]s in that category. It is a [[quadratic operad]] whose [[Koszul dual]] is the *[[commutative operad|operad for commutative algebras]]*.

(This [[Koszul duality]] is what makes [[L-∞ algebras]] be equivalent to (semifree) [[differential graded coalgebras]].)

In the case of the monoidal category of chain complexes, we also say dg-Lie-operad. Its cofibrant [[resolution]] in the [[model structure on operads]] is the [[L-infinity operad]] whose [[algebras over an operad]] are [[L-∞-algebra]]s.

## Related concepts

* [[associative operad]]

* [[commutative operad]]

* **Lie operad**

## References

* [[Todd Trimble]], _[[toddtrimble:Notes on operads and the Lie operad]]_ ([pdf](http://math.ucr.edu/home/baez/trimble/trimble_lie_operad.pdf))

* V. Hinich, _Homological algebra of homotopy algebras_,  Communications in algebra, 25(10) (1997), 3291-3323, [pdf](http://math.haifa.ac.il/hinich/WEB/mypapers/haha.pdf); erratum [math.AG/0309453](http://arxiv.org/abs/math.AG/0309453)

[[!redirects Lie algebra operad]]


Lie theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
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## Classical Lie theory and beyond

__Lie theory__ (wikipedia: [Lie theory](http://en.wikipedia.org/wiki/Lie_theory)) is the study of [[Lie groups]], [[Lie algebras]], their actions as groups of transformations, their representations, their
cousins ([[Chevalley group]]s, [[quantum group]]s, Banach Lie groups, [[Lie bialgebra]]s, [[universal enveloping algebra]]s)
and categorifications (higher Lie groups, $L_\infty$-[[L-infinity-algebra|algebras]] etc.), including horizontal: [[Lie groupoid]], [[Lie algebroid]], $L_\infty$-[[L-infinity-algebroid|algebroid]].

[[Lie group]] is a [[group|groups]] [[internalization|internal to]] [[Diff]]. The [[Lie algebras]] over real or complex numbers appear as the linearization of real or complex Lie groups; the infinitesimal version of Lie groups are [[local Lie group]]s. The relation between real Lie algebras and real Lie groups was established by [[Lie's three theorems]] and the study of this correspondence is the classical Lie theory in narrow sense.

In a categorical context there are the usual two generalization of Lie groups: the [[horizontal categorification]] -- or oidification -- which adds _more objects_ and leads to _Lie groupoids_; and the [[vertical categorification]] which adds _higher morphisms_ and leads to $\infty$-Lie groups and $\infty$-Lie groupoids. This is described below.

In the course of these categorifications one usually finds that working [[internalization|internal to]] $Diff$ is too restrictive for many purposes. Therefore higher Lie theory is often considered internal to [[generalized smooth space|generalized smooth spaces]].

## Lie groupoids versus Lie algebroids

[[horizontal categorification|Oidification]] leads from Lie groups to [[Lie groupoid|Lie groupoids]]
and from [[Lie algebra|Lie algebras]] to
[[Lie algebroid|Lie algebroids]].

When it was found that Lie algebroids integrate to Lie groupoids by mapping paths into them, not only a "well known" but apparently also well forgotten way to integrate Lie algebras by means of paths was rediscovered, but also the idea to integrate higher Lie algebroids by mapping higher dimenmsional paths into them was clearly suggested.

## Higher Lie theory: $\infty$-Lie groupoids and $L_\infty$-algebroids

bla bla bla
[[infinity-groupoid]] bla bla bla
[[Lie infinity-groupoid]] bla bla bla
[[Lie infinity-algebroid]] bla bla bla

## Related entries

Smooth spaces and smooth groups: [[homogeneous space]], [[Lie group]], [[Lie groupoid]], [[Morita equivalence]] of Lie groupoids, [[NQ-supermanifold]], [[orbifold]], [[differentiable stack]], [[generalized smooth algebra]], [[generalized smooth space]], [[Lie infinity-groupoid]], [[local Lie group]], [[Poisson-Lie group]], [[symmetric space]]

Algebraic cousins and tools: [[algebraic group]]
[[Chevalley-Eilenberg algebra]], [[CoDGCA]], [[coalgebra]], [[L-infinity-algebra]], [[Lie algebra]], [[Lie algebroid]], [[Lie infinity-algebroid]],
[[universal enveloping algebra]], [[PBW theorem]], [[Ado's theorem]], [[Hausdorff series]], [[Duflo isomorphism]], [[coexponential map]], [[free graded co-commutative coalgebra]], [[quantum group]], [[Chevalley group]], [[Lie bialgebra]], [[Leibniz algebra]]

Geometry on Lie groups and related spaces: [[integration over supermanifolds]], [[space and quantity]], [[superdifferential form]], [[volume of a Lie groupoid]], [[Maurer-Cartan form]], [[invariant vector field]]

Lie theory in narrow sense (Lie group(oid)s vs. algebr(oid)s): [[tangent Lie algebra]], [[tangent Lie algebroid]], [[Lie integration]], [[Lie theory for stacky Lie groupoids]], [[Lie's three theorems]], [[exponential map]]

Applications where Lie objects appear: [[matrix theory]], [[connection on a bundle]], [[BRST complex]], [[BV theory]],  [[examples for Lagrangian BV]], [[fundamental groupoid]], [[geometric quantization]], [[invariant theory]], [[representation theory]], [[conformal field theory]]

There are also many special kinds of [[Lie groups]] ([[semisimple Lie group|semisimple]], solvable, nilpotent, [[general linear group|general linear]], [[special linear group|special linear]], [[metaplectic group]], [[unitary group]] etc.), kinds of [[Lie algebras]] ([[Kac-Moody algebra|Kac-Moody]], [[semisimple Lie algebra|semisimple]], [[affine Lie algebra|affine]] Lie algebras, [[Virasoro Lie algebra]]...), structure notions on Lie groups and algebras (maximal torus, [[Killing form]], Dynkin diagram, root system, Cartan subalgebra, Gauss decomposition, Bruhat decomposition...),  and versions and tools of [[representation theory]] ([[Young diagram]], [[symmetric function]], [[Schur function]], [[combinatorial representation theory]], [[geometric representation theory]], weight lattice, [[induced representation]], [[Verma module]], [[Langlands program]]), see those entries.

## References

### General

* {#Onishchik93} A. L. Onishchik (ed.) _Lie Groups and Lie Algebras_

  * *I.* A. L. Onishchik, E. B.  Vinberg, _Foundations of Lie Theory_,

  * *II.* V. V. Gorbatsevich, A. L. Onishchik, _Lie Transformation Groups_

  Encyclopaedia of Mathematical Sciences, Volume 20, Springer 1993

* {#HilgertNeeb12} Joachim Hilgert, [[Karl-Hermann Neeb]], _Structure and Geometry of Lie Groups_, Springer Monographs in Mathematics, Springer-Verlag New York, 2012 ([doi:10.1007/978-0-387-84794-8](https://link.springer.com/book/10.1007/978-0-387-84794-8))

* {#Okounkov18} [[Andrei Okounkov]], _New worlds for Lie theory_, talk at [ICM 2018](http://www.icm2018.org/portal/en/home) ([pdf](http://www.math.columbia.edu/~okounkov/icm.pdf))

### Lie 1-groupoids vs. Lie algebroids

The study of Lie theory of Lie groupoids and Lie algebroids was notably motivated by Alan Weinstein's studies of [[geometric quantization]] of [[Poisson manifold|Poisson manifolds]], as well as by [[Kirill Mackenzie|Kirill Mackenzie's]] work.

The canonical review for the integration theory of Lie algebroids (and hence for Lie algebras by the **path integration method**) is:

* Marius Crainic, Rui Loja Fernandes, _Lectures on Integrability of Lie Brackets_, ([arXiv](http://aps.arxiv.org/abs/math.DG/0611259),[blog](http://golem.ph.utexas.edu/category/2008/05/integrability_of_lie_brackets.html)).

This focuses on integration to [[Lie groupoid|Lie groupoids]] proper, i.e. to integration internal to manifolds. In contrast to Lie algebras, not every Lie algebroid integrates to such a proper [[Lie groupoid]], though: the space of morphisms of the Lie groupoid is a quotient of paths in the Lie algebroid by Lie algebroid homotopies, and this quotient may not exist as a manifold. (Crainic and Fernandes [discuss](http://aps.arxiv.org/abs/math.DG/0611259) the obstruction in detail.) But the quotient of course always exists as a [[weak quotient]] or [[homotopy quotient]] or [[stacky quotient]] itself. The result of realizing the space of morphisms of the integrated Lie algebroid as such a stacky quotient has been studied by [Chenchang Zhu](http://chenchangzhu.blogspot.com/):
[[Lie theory for stacky Lie groupoids]]

### Higher categorical

See at _[[Lie integration]]_.

[[!redirects Lie Theory]]

category: Lie theory


Lie Theory > history

< [[Lie Theory]]
[[!redirects Lie Theory -- history]]


Lie theory for stacky Lie groupoids

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## Definition: Stacky Lie groupoid

A _stacky Lie groupoid_ (in contrast to a [[Lie groupoid]]) is an [[internal category|internal groupoid]] in the category of [[differentiable stack|differentiable stacks]], such that the space of objects is an ordinary [[manifold]].

Notice that differentiable stacks are equivalent to Lie groupoids modulo [[homotopy theory|Morita equivalence]].

## Literature

Stacky Lie groupoids and their [[Lie theory]] were introduced and studied by [[Chenchang Zhu]] and collaborators.

* [[Chenchang Zhu]], _n-Groupoids and Stacky Groupoids_, 2008, International Mathematics Research Notices (2009) 2009:4087-4141; (http://arxiv.org/abs/0801.2057 arXiv:0801.2057). DOI: (http://dx.doi.org/10.1093/imrn/rnp080 10.1093/imrn/rnp080).

* C.Z., _Lie $n$-groupoids and stacky Lie groupoids_ ([arXiv](http://arxiv.org/abs/math/0609420))

* C. Z.,  _Lie II theorem for Lie algebroids via stacky Lie groupoids_ ([arXiv](http://arxiv.org/abs/math/0701024))

* C. Z. _Lie II theorem_ ([pdf slides](http://cib.epfl.ch/hosted/programs/poisson2008/cms/fileadmin/misc/lie2talk2.pdf))

* [[Henrique Bursztyn]], C.Z., _Morita equivalence of Poisson manifold via stack groupoids_ ([arXiv](http://arxiv.org/abs/0707.2175))

## Further resources

* $n$-Café blog discussion about this is [here](http://golem.ph.utexas.edu/category/2008/12/zhu_on_lies_second_theorem_for.html).


Lie triple system

* William G. Lister, _A structure theory of Lie triple systems_, Trans. Amer. Math. Soc. __72__ (1952), 217-242, [doi](http://dx,doi.org/10.1090/S0002-9947-1952-0045702-9), [pdf](http://www.ams.org/journals/tran/1952-072-02/S0002-9947-1952-0045702-9/S0002-9947-1952-0045702-9.pdf)
* Florian Girelli, _[[Snyder space]]-time: K-loop and Lie triple system_, SIGMA Symmetry Integrability Geom. Methods Appl. __6__ (2010), Paper 074, 19 pp. [journal](http://www.emis.de/journals/SIGMA/2010/074) [pdf](http://www.emis.de/journals/SIGMA/2010/074/sigma10-074.pdf) [arxiv/1009.4762](http://arxiv.org/abs/1009.4762)


Lie's three theorems

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## 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](http://www.encyclopediaofmath.org/index.php/Lie_theorem) 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 space|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](https://doi.org/10.1007/978-3-642-56936-4)]

+-- {: .un_remark}
###### 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 groupoid|Lie groupoids]]. In other words, there are analogous properties for the differentiation functor
$$diff : LieGroupoids \to LieAlgebroids.$$
from [[Lie groupoid|Lie groupoids]] to [[Lie algebroid|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:

* [[Marius Crainic]], [[Rui Loja Fernandes]], *Integrability of Lie brackets*, Ann. of Math. **157** 2 (2003) 575-620 [[arXiv:math.DG/0105033](http://arxiv.org/abs/math.DG/0105033), [doi:10.4007/annals.2003.157.575](https://doi.org/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](https://doi.org/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 functor|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](https://arxiv.org/abs/math/0611259), [doi:10.2140/gtm.2011.17.1](http://dx.doi.org/10.2140/gtm.2011.17.1)]

and in the introduction of

* [[Chenchang Zhu]], _Lie II theorem for Lie algebroids via stacky Lie groupoids_, ([arXiv](http://arxiv.org/abs/math/0701024))

## Related concepts

* [Lie's third theorem for Leibniz algebras](Leibniz+algebra#LieThirdTheorem)

[[!redirects Lie's theorem]]
[[!redirects Lie's theorems]]
[[!redirects Lie's first theorem]]
[[!redirects Lie's second theorem]]
[[!redirects Lie's third theorem]]


Lie-Poisson structure

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## Idea

For $\mathfrak{g}$ a [[Lie algebra]], the [[underlying]] [[dual vector space]] $\mathfrak{g}^*$ canonically inherits the structure of a [[Poisson manifold]] whose Poisson [[Lie bracket]] reduces on [[linear functions]] $\mathfrak{g} \hookrightarrow C^\infty(\mathfrak{g}^*)$ to the original [[Lie bracket]] on $\mathfrak{g}$. This is the **Lie-Poisson structure** on $\mathfrak{g}^*$.

More generally, for $\mathfrak{a}$ a [[Lie algebroid]] the fiberwise dual $\mathfrak{a}^*$ inherits such a Poisson manifold structure.

[[Poisson manifold]] structures of this form are also called _[[linear Poisson structures]]_.

## Definition

### Abstractly

First notice that for $f \in C^\infty(\mathfrak{g}^\ast)$ as smooth function on the dual of a Lie algebra, then its [[de Rham differential]] 1-form at some $\alpha \in \mathfrak{g}^\ast$, being a linear map

$$
  \mathbf{d} f|_{\alpha}
  \colon
  T_\alpha \mathfrak{g}^\ast
  =
  \mathfrak{g}^\ast
  \longrightarrow
  \mathbb{R}
$$

is canonically identified with a Lie algebra element itself.

With this understood, then for $f,g \in C^\infty(\mathfrak{g}^*)$ two [[smooth functions]] on $\mathfrak{g}^*$ their Poisson [[Lie bracket]] in the Lie-Poisson structure is defined by

$$
  \{f,g\} \;\colon\; \theta \mapsto -\theta ([\mathbf{d} f, \mathbf{d} g])
  \,.
$$

Notice that for $v\in \mathfrak{g}$ regarded as a linear function $\langle -,v\rangle$ on $\mathfrak{g}^\ast$, then under the above identification we have $\mathbf{d} \langle -,v\rangle = v$. This means that on linear functions the Lie-Poisson bracket is simply the original Lie bracket:

$$
  \left\{
    \langle -, v_1\rangle,
    \langle -, v_2\rangle,
  \right\}
  =
  \langle - ,[v_1,v_2]\rangle
  \,.
$$

This Lie-Poisson structure may be thought of as the unique smooth extension of this bracket on linear functions to all smooth functions on $\mathfrak{g}^\ast$.

### In components
 {#DefinitionInComponents}

Let $\{x^a\}$ be a [[basis]] for the [[vector space]] underlying the given [[Lie algebra]] $\mathfrak{g}$. Write $\{C^{a b}{}_c\}$  for the components of the [[Lie bracket]] $[-,-]$ in this basis (the structure constants), given by

$$
  [x^a,x^b] = \underset{c}{\sum} C^{a b}{}_c x^c
  \,.
$$

Write $\{\partial_a\}$ for the dual basis of the [[dual vector space]] $\mathfrak{g}^\ast$, so that the pairing $\mathfrak{g}^\ast \otimes\mathfrak{g} \to \mathbb{R}$ is given by

$$
  \partial_a x^b = \delta_a^b =
  \left\{
    \array{
      1 & if\; a=b
      \\
      0 & otherwise
    }
  \right.
$$

As the notation is meant to suggest, dually the $\{x^a\}$ may be regarded as basis for the linear functions on $\mathfrak{g}^\ast$ and the $\{\partial_a\}$ serve as a basis of [[vector fields]] on $\mathfrak{g}^\ast$.

With this identification understood, the [[multivector fields]] on $\mathfrak{g}^\ast$ are spanned by elements of the form

$$
  v^{a_1 \cdots a_q} \partial_{a_1}\wedge \cdots \wedge \partial_{a_q}
$$

(with the sum over indices understood) for $\{v^{a_1 \cdots a_q}\}$ smooth functions on $\mathfrak{g}^\ast$.

The [[Poisson tensor]] $\pi \in \wedge^2 \Gamma(T\mathfrak{g}^\ast)$ of the Lie-Poisson structure is given by

$$
  \pi = \tfrac{1}{2}\underset{a,b,c}{\sum} C^{a b}{}_c x^c \partial_a \wedge \partial_b
  \,.
$$

The [[Schouten bracket]] on multivector fields is given on linear basis elements by

$$
  \{x^a, x^b\}_{Sch} = 0
$$

$$
  \{\partial_a, x^b\}_{Sch} = \delta_a^b
$$

$$
  \{\partial_a, \partial_b\}_{Sch} = 0
$$

(the [[canonical commutation relations]]) and extended as a graded [[derivation]] in both arguments.

## Properties

### Deformation quantization by universal enveloping algebra

See at _[[formal deformation quantization]]_ the section _[Relation to universal enveloping algebras](formal+deformation+quantization#RelationToUniversalEnvelopingAlgebras)_.

### Symplectic groupoid

The [[symplectic groupoid]] [[Lie integration|integrating]] the Lie-Poisson structure on $\mathfrak{g}^*$ is the [[action groupoid]] $\mathfrak{g}^* \sslash G$ of the [[coadjoint action]]. For more see at _[[symplectic groupoid]]_ in the section _[Examples -- Of Lie-Poisson stucture](symplectic+groupoid#OfLiePoissonStructure)_.

### Symplectic leaves

The [[symplectic leaves]] of the Lie-Poisson structure on $\mathfrak{g}^*$ are the [[coadjoint orbits]].

### Poisson-Lie algebroid cohomology
 {#PoissonLieAlgebroidCohomology}

We consider the [[Poisson Lie algebroid]] $\mathfrak{P}(\mathfrak{g}^\ast)$ of a Lie-Poisson structure and the [[Lie algebroid cohomology]].

+-- {: .num_remark}
###### Remark

By the discussion at _[[Poisson Lie algebroid]]_, the graded algebra of [[multivector fields]] equipped with the [[differential]] given by the [[Schouten bracket]] with the [[Poisson bivector]]

$$
  d_{CE} = \{\pi, -\}_{Sch}
$$

is the [[Chevalley-Eilenberg algebra]] of this Lie algebroid:

$$
  CE(\mathfrak{P}(\mathfrak{g}^\ast))
  =
  \left(
    \wedge^\bullet \Gamma(T\mathfrak{g}^\ast),
    d_{CE} = \{\pi, -\}_{Sch}
  \right)
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

As for every [[Poisson Lie algebroid]], the [[Poisson bivector]]
$\pi \in CE(\mathfrak{P}(\mathfrak{g}^\ast))$ is a [[Lie algebroid cocycle]] of degree 2

$$
  d_{CE}\pi = \{\pi,\pi\}_{Sch} = 0
$$

(see also at _[[symplectic Lie n-algebroid]]_).

In view of the fact that here $\pi$ is just another incarnation of the [[Lie bracket]], this condition here is an incarnation of the [[Jacobi identity]] on the Lie algebra $(\mathfrak{g},[-,-])$.

=--

But in the simple case of Lie-Poisson structure, this cocycle is in fact exact:

+-- {: .num_prop #PoissonTensorCoboundary}
###### Proposition

For the Poisson-Lie structure on $\mathfrak{g}^\ast$ the [[Poisson tensor]] $\pi \in CE^2(\mathfrak{P}(\mathfrak{g}))$ has a [[coboundary]] and hence is trivial in [[Lie algebroid cohomology]].

=--

+-- {: .proof}
###### Proof

Consider the component-description from [above](#DefinitionInComponents).
We show that $x^a \partial_a$ is a coboundary.

First notice that

$$
  \{x^b, x^a \partial_a \}_{Sch} = -x^b
$$

and

$$
  \{\partial_b, x^a \partial_a \}_{Sch} = \partial_b
  \,.
$$

From this we get

$$
  \begin{aligned}
    d_{CE} (x^a \partial_a)
    &=
   \left\{\pi,\;  x^a \partial_a\right\}_{Sch}
    \\
    & =
    \left\{\frac{1}{2} C^{a b}{}_c x^c \partial_a \wedge \partial_b,\;
    x^a \partial_a\right\}_{Sch}
    \\
    & = (2-1) \frac{1}{2} C^{a b}{}_c x^c \partial_a \wedge \partial_b
    \\
    & =
    \pi
  \end{aligned}
$$

=--

### Poisson Lie group structure

Under addition a Lie-Poisson manifold becomes a _[[Poisson Lie group]]_, see there for more.

## Related concepts

* a _[[moment map]]_ is often expresses as a Poisson homomorphism into a Lie-Poisson structure.

## References

The notion of Lie-Poisson structures was originally found by [[Sophus Lie]] and then rediscovered by [[Felix Berezin]] and by [[Alexander Kirillov]], [[Bertram Kostant]] and [[Jean-Marie Souriau]].

General accounts:

* {#BayenFlatoFronsdalLichnerowiczSternheimer78} [[François Bayen]], [[Moshé Flato]], [[Christian Fronsdal]], [[André Lichnerowicz]], [[Daniel Sternheimer]], §2 in: _Deformation theory and quantization. I. Deformations of symplectic structures._, Annals of Physics **111** 1 (1978) 61-110 [doi:10.1016/0003-4916(78)90224-5]

* [[Alan Weinstein]], §3 in: *The local structure of Poisson manifolds*,  J. Differential Geom. **18** 3 (1983) 523-557 [[doi:10.4310/jdg/1214437787](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-18/issue-3/The-local-structure-of-Poisson-manifolds/10.4310/jdg/1214437787.full)]

*  Izu Vaisman, section 3.1 of _Lectures on the Geometry of Poisson Manifolds_, Birkhäuser (1994)

* [[Camille Laurent-Gengoux]], _Linear Poisson Structures and Lie Algebras_, chapter 7 of: Camille Laurent-Gengoux, Anne Pichereau, [[Pol Vanhaecke]] (eds.), _Poisson Structures_, Grundlehren der mathematischen Wissenschaften **347** 179-203 [[doi:10.1007/978-3-642-31090-4_7](https://doi.org/10.1007/978-3-642-31090-4_7)]

Monograph:

* {#Waldmann07} [[Stefan Waldmann]], pp. 220 in: *Poisson-Geometrie und Deformationsquantisierung*, Springer (2007) [[doi:10.1007/978-3-540-72518-3](https://doi.org/10.1007/978-3-540-72518-3)]

On the [[formal deformation quantization]] of Lie-Poisson structures via transfer of the product on the [[universal enveloping algebra]] of the given Lie algebra:

* {#Gutt83} [[Simone Gutt]], *An explicit $\ast$-product on the cotangent bundle of a Lie group*, Letters in Mathematical Physics **7** (1983) 249–258 [[doi:10.1007/BF00400441](https://doi.org/10.1007/BF00400441)]

reviewed in:

* {#Gutt11} [[Simone Gutt]], §2.2 of: _Deformation quantization of Poisson manifolds_, Geometry and Topology Monographs **17** (2011) 171-220 [[gtm:17](https://msp.org/gtm/2011/17/p003.xhtml), [pdf](http://msp.org/gtm/2011/17/gtm-2011-17-003p.pdf), [arXiv:math/0003107](https://arxiv.org/abs/math/0003107)]

and generalization to more general polynomial Poisson algebras is discussed in

* Michael Penkava, Pol Vanhaecke, _Deformation Quantization of Polynomial Poisson Algebras_, Journal of Algebra
227, 365ñ393 (2000) ([arXiv:math/9804022](https://arxiv.org/abs/math/9804022))

On [[strict deformation quantization]] of [[Lie-Poisson structures]] via [[Lie group|Lie]]-[[group algebras]]:

* [[Marc A. Rieffel]], *Lie Group Convolution Algebras as Deformation Quantizations of Linear Poisson Structures*, American Journal of Mathematics **112** 4 (1990) 657-685 [[doi:10.2307/2374874](https://doi.org/10.2307/2374874), [jstor:2374874](https://www.jstor.org/stable/2374874)]

The [[symplectic Lie groupoid]] [[Lie integration|Lie integrating]] Lie-Poisson structures is discussed as example 4.3 in:

* {#BursztynCrainic} [[Henrique Bursztyn]], [[Marius Crainic]], _Dirac structures, momentum maps and quasi-Poisson manifolds_ ([pdf](http://www.preprint.impa.br/FullText/Bursztyn__Fri_Dec_23_11_24_19_BRDT_2005.html/alanfestimpa.pdf))

See also

* [[Victor Ginzburg]], [[Alan Weinstein]], _Lie-Poisson structure on some Poisson Lie groups_, Journal of the AMS, volume 5, number 2 (1992) ([pdf](http://www.ams.org/journals/jams/1992-05-02/S0894-0347-1992-1126117-8/S0894-0347-1992-1126117-8.pdf))

[[!redirects Lie-Poisson structures]]

[[!redirects Lie-Poisson manifold]]
[[!redirects Lie-Poisson manifolds]]


Lie-Rinehart pair

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## Idea

The notion of _Lie--Rinehart pair_ is an algebraic encoding of the notion of _[[Lie algebroid]]_.  It is the pair consisting of the [[associative algebra]] of [[functions]] on the base space of the Lie algebroid and of the [[Lie algebra]] of its global [[sections]]. The [[anchor map]] of the Lie algebroid is encoded in the [[action]] of the Lie algebra on the associative algebra by [[derivations]] and the local structure is encoded in the Lie algebra being a [[module]] over the associative algebra.

Since in this formulation the base manifold of the Lie algebroid is entirely described [[Isbell duality|dually]] in terms of its [[algebra of functions]], and since the definition does not refer to this being a _commutative_ algebra, the notion of Lie-Rinehart pair in fact generalizes the notion of Lie algebroid from ordinary [[differential geometry]] to [[noncommutative geometry]].

## Definition

A _Lie--Rinehart-pair_ $(A,\mathfrak{g})$ is a pair consisting of

1. an [[associative algebra]] $A$

1. a [[Lie algebra]] $\mathfrak{g}$

such that

1. $A$ is a $\mathfrak{g}$-[[module]]

1. $\mathfrak{g}$ is an $A$-[[module]]

with both module structures being compatible in the obvious way:

1. $\mathfrak{g}$ acts as [[derivations]] of $A$: that is, we have a Lie algebra [[homomorphism]] $\mathfrak{g} \to Der(A)$.

1. $A$ acts as linear transformations of $\mathfrak{g}$ in a way obeying the [[Leibniz rule]]: that is, we have an associative algebra homomorphism $A \to End(\mathfrak{g})$, where $End(\mathfrak{g})$ is the algebra of all linear transformations of $\mathfrak{g}$, such that
$$  [v, a w] = v(a) w + a [v,w]. $$

## Examples

In the case that $A = C^\infty(X)$ is the algebra of smooth functions on a smooth [[manifold]] $X$, Lie--Rinehart pairs $(C^\infty(X), \mathfrak{g})$ are naturally identified with [[Lie algebroid]]s over $X$: given the [[Lie algebroid]] in its incarnation as a [[vector bundle]] morphism

$$
  \array{
     E &&\stackrel{\rho}{\to}&& T X
     \\
     & \searrow && \swarrow
     \\
     && X
  }
$$

equipped with a bracket

$$
  [-,-] : \Gamma(E) \otimes\Gamma(E) \to \Gamma(E)
$$

we obtain a Lie--Rinehart pair by setting

* $\mathfrak{g} = \Gamma(E)$ is the [[Lie algebra]] of [[sections]] of $E$ using the above bracket

* the action of $A$ on $\mathfrak{g}$ is the obvious multiplication of sections of vector bundles over $X$ by functions on $X$

* the action of $\mathfrak{g}$ on $C^\infty(X)$ is given by first applying the anchor map $\rho$ and then using the canonical action of vector fields on functions.

So for all the examples listed at [[Lie algebroid]] we obtain an example for Lie--Rinehart pairs.

In particular

* the Lie--Rinehart pair coresponding to the [[tangent Lie algebroid]] of a [[manifold]] $X$ is $(C^\infty(X), \Gamma(T X))$ with the obvious action on each other.

* the Lie--Rinehart pair corresponding to an ordinary [[Lie algebra]] $\mathfrak{g}$ is $(\mathbb{R}, \mathfrak{g})$ with $\mathfrak{g}$ acting trivially on $\mathbb{R}$.

* the Lie--Rinehart pair corresponding to a [[Poisson Lie algebroid]] on a [[Poisson manifold]] $X$ is $(C^\infty(X), MultVect(X))$, where the [[Lie algebra]] is the the space of [[multivector fields]] on $X$ equipped with the [[Schouten bracket]].

In [open-closed string field theory](string%20field%20theory#OpenClosedStringFieldTheory) one finds at least one half of the axioms of homotopy Lie-Rinehart pairs.

## Generalizations

A little bit is known in the literature to generalizations of the notion of Lie--Rinehart algebras that are to [[Lie ∞-algebroids]] as the latter are to [[Lie algebroids]].

In

* [[Dmitry Roytenberg]], _Courant--Dorfman algebras and their cohomology_ ([arXiv](http://arxiv.org/abs/0902.4862))

the analogous algebraic structure for [[Courant algebroid]]s is discussed. These "2-Lie--Rinehart algebras" are called [[Courant-Dorfman algebra]]s there.

## Related concepts

* [[bisections of a Lie groupoid]]

## References

The original reference:

* G. Rinehart, _Differential forms for general commutative algebras_, Trans. Amer. Math. Soc. __108__ (1963), 195-222

On the history of this and related concepts:

* {#Huebschmann22} [[Johannes Huebschmann]], *On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras* [[arXiv:2208.02539](https://arxiv.org/abs/2208.02539)]

Review:

* [[Johannes Huebschmann]], Sec 1 of: _Lie--Rinehart algebras, descent, and quantization_ ([arXiv](http://arxiv.org/abs/math/0303016))

* [[Mikhail Kapranov|M. Kapranov]], _Free Lie algebroids and the space of paths_,	Sel. Math. (N.S.) __13__, n. 2 277--319 (2007), [arXiv:math.AG/0702584](http://arxiv.org/abs/math.AG/0702584), [doi](http://dx.doi.org/10.1007/s00029-007-0041-9)

* V. Nistor, [[Alan Weinstein]], [[Ping Xu]], _Pseudodifferential operators on differential groupoids_, Pacific J. Math. __189__, 117–152 (1999)

A notion of universal enveloping algebra of a Lie--Rinehart algebra is discussed in

* [[Ieke Moerdijk]], [[Janez Mrčun]], _On the universal enveloping algebra of a Lie--Rinehart algebra_, Proc. Amer. Math. Soc. in press, ([pdf](http://www.ams.org/proc/0000-000-00/S0002-9939-10-10347-5/S0002-9939-10-10347-5.pdf), [arXiv/0801.3929](http://arxiv.org/abs/0801.3929))

A connection with [[BV-theory]] and [[L-infinity algebra]] is made in

* {#Kjeseth01} [[Lars Kjeseth]], _Homotopy Rinehart cohomology of homotopy Lie-Rinehart pairs_, Homology Homotopy Appl. Volume 3, Number 1 (2001), 139-163. ([Euclid](https://projecteuclid.org/euclid.hha/1140370269))

* [[Johannes Huebschmann]], _Lie--Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras_ ([journal](http://aif.cedram.org/item?id=AIF_1998__48_2_425_0))

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[[!redirects Lie-Rinehart algebra]]
[[!redirects Lie?Rinehart pair]]
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[[!redirects Lie--Rinehart pair]]
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LieAlg

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## Definition

The [[category]] **$Lie Alg$** is that whose [[object]]s are [[Lie algebra]]s $(\mathfrak{g}, [-,-]_{\mathfrak{g}})$ and whose morphisms are Lie algebra [[homomorphisms]], that is linear maps $\phi\colon \mathfrak{g} \to \mathfrak{h}$ such that for all $x,y \in \mathfrak{g}$ we have

$$
  \phi( [x,y]_{\mathfrak{g}}) = [\phi(x),\phi(y)]_\mathfrak{h}
  \,.
$$

If Lie algebras are expressed in terms of their [[Chevalley–Eilenberg algebra]]s (and if restricted to finite-dimensional Lie algebras), this may equivalently be characterized as follows:

$Lie Alg$ is the [[full subcategory]] of the [[opposite category]] of the category [[dgAlg]] of [[dg-algebras]] on those dg-algebras whose underlying [[graded algebra]] is a [[Grassmann algebra]], i.e. of the form $\wedge^\bullet \mathfrak{g}$.

## Special objects

* [[abelian Lie algebra]]

* [[simple Lie algebra]]

* [[semisimple Lie algebra]]

category: category

[[!redirects LieAlg]]
[[!redirects Lie Alg]]


LieGrp

The [[category]] of [[Lie group]]s and smooth group homomorphisms.


LieGrpd

The [[category]] of [[Lie groupoids]] and smooth [[functors]] between them.

The [[localization]] of this at the equivalence-[[Morita morphisms]] is equivalently the category of [[differentiable stacks]].

category: category


Lieven Le Bruyn

__Lieven Le Bruyn__ is a mathematician at [Universiteit Antwerpen](http://www.ua.ac.be).

* [web](http://win.ua.ac.be/~lebruyn)

* blog: [neverendingbooks](http://www.neverendingbooks.org)

category:people

[[!redirects Lieven LeBruyn]]


Lifschitz realizability

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## Idea

(...)

## Related concepts

* [[realizability]]

## References

* [[Vladimir Lifschitz]], *$CT_0$ is stronger than $CT_0!$*. Proceedings of the American Mathematical Society, Volume 73, Number 1, January 1979, pp. 101–106. [[pdf](https://www.ams.org/journals/proc/1979-073-01/S0002-9939-1979-0512067-X/S0002-9939-1979-0512067-X.pdf)]

* [[Jaap van Oosten]], *Lifschitz realizability*. The Journal of Symbolic Logic, Volume 55, Issue 2 (1990), pp. 805-821. [[doi:10.2307/2274666](https://doi.org/10.2307/2274666)]

* [[Jaap van Oosten]], *Two remarks on the Lifschitz realizability topos*. The Journal of Symbolic Logic, Volume 61, Issue 1, March 1996, pp. 70-79. [[doi:10.2307/2275598](https://doi.org/10.2307/2275598)]

* [[Ray-Ming Chen]], [[Michael Rathjen]], *Lifschitz realizability for intuitionistic Zermelo-Fraenkel set theory*. Archive for Mathematical Logic, Volume 51 (2012), pp. 789–818. [[doi:10.1007/s00153-012-0299-2](https://doi.org/10.1007/s00153-012-0299-2), [pdf](https://www1.maths.leeds.ac.uk/~rathjen/Lifschitz.pdf)]

* [[Michael Rathjen]], [[Andrew Swan]], *Lifschitz Realizability as a Topological Construction*. The Journal of Symbolic Logic, Volume 85, Issue 4, December 2020, pp. 1342 - 1375. [[doi:10.1017/jsl.2021.1](https://doi.org/10.1017/jsl.2021.1), [arXiv:1806.10047](https://arxiv.org/abs/1806.10047)]

* Takayuki Kihara, *Degrees of incomputability, realizability and constructive reverse mathematics* [[arXiv:2002.10712](https://arxiv.org/abs/2002.10712)]