Lie group

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###Context###
#### Group Theory
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[[!include group theory - contents]]
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#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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#### $\infty$-Lie theory
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[[!include infinity-Lie theory - contents]]
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# Contents
* table of contents
{:toc}

## Idea

A **Lie group** is a [[group]] with [[smooth structure]]. Lie groups form a category, [[LieGrp]].

## Definition

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

A **Lie group** is a [[smooth manifold]] whose [[forgetful functor|underlying]] [[set]] of [[elements]] is equipped with the [[structure]] of a [[group]] such that the [[magma|group multiplication]] and [[inverse]]-assigning [[functions]] are [[smooth functions]].

In other words, a Lie group is a [[group object]] [[internalization|internal to]] the [[category]] [[SmthMfd]] of [[smooth manifolds]].

=--

Usually the [[smooth manifold]] is assumed to be defined over the [[real numbers]] and to be of [[finite number|finite]] [[dimension]] (f.d.), but extensions of the definition to some other [[ground fields]]  or to -[[infinite-dimensional manifolds]] are also relevant, sometimes under other names (such as [[Fréchet Lie group]] when the underlying manifold is an infinite-dimensional [[Fréchet manifold]]).

A real Lie group is called a _[[compact Lie group]]_ (or _connected_, _simply connected_ Lie group, etc) if its underlying [[topological space]] is [[compact space|compact]] (or [[connected space|connected]], [[simply connected space|simply connected]], etc).

Every connected finite dimensional real Lie group is [[homeomorphism|homeomorphic]] to a [[product]] of a [[compact Lie group]] (its [[maximal compact subgroup]]) and a [[Euclidean space]].

Every [[abelian group|abelian]] connected compact finite dimensional real Lie group is a [[torus]] (a product of [[circles]] $T^n = S^1\times S^1 \times \ldots \times S^1$).

There is an [[infinitesimal space|infinitesimal]] version of a Lie group, a so-called [[local Lie group]], where the multiplication and the inverse are only partially defined, namely if the arguments of these operations are in a sufficiently small neighborhood of identity. There is a natural equivalence of local Lie groups by means of agreeing (topologically and algebraically) on a smaller neighborhood of the identity. The category of local Lie groups is equivalent to the category of connected and simply connected Lie groups.

The first order [[infinitesimal object|infinitesimal]] approximation to a Lie group is its [[Lie algebra]].

## Properties

### Lie's three theorems

[[Sophus Lie]] proved several theorems, known as [[Lie's three theorems]], on the relationship between [[Lie algebras]] and Lie groups. Lie's third theorem is about the [[equivalence of categories]] of finite-dimensional real Lie algebras and local Lie groups. Because [[Élie Cartan]] extended this to a global integrability theorem, Lie's third theorem is also called the Cartan-Lie theorem.

### Lie subgroups

\begin{prop}\label{CartanClosedSubgroupTheorem}
**([[Cartan's closed subgroup theorem]])**

If $H \subset G$ is a [[closed subgroup]] of a ([[finite number|finite]] [[dimension of a manifold|dimensional]]) [[Lie group]], then $H$ is a sub-Lie group, hence a [[smooth manifold|smooth]] [[submanifold]] such that its group operations are [[smooth functions]] with respect to the the [[submanifold]] [[smooth structure]].
\end{prop}

### Classification

\begin{prop}

Every connected finite-dimensional real Lie group is [[homeomorphism|homeomorphic]] to a [[product]] of a [[compact Lie group]] and a [[Cartesian space|Euclidean space]]. Every abelian connected compact f.d. real Lie group is a [[torus]] (a product of circles $T^n = S^1\times S^1 \times \ldots \times S^1$).

\end{prop}

The [[simple Lie group]]s have a classification into infinite series of

* [[classical Lie group]]s

and a finite snumber o

* [[exceptional Lie group]]s

### Different Lie group structures on a group

For $G$ a bare [[group]] (without smooth structure) there may be more than one way to equip it with the [[stuff, structure, property|structure]] of a Lie group.

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

As bare [[abelian group]]s, the  [[Cartesian space]]s $\mathbb{R}^n$ are, for all $n$, [[vector space]]s over the [[rational number]]s $\mathbb{Q}$ whose [[dimension]] is the [[cardinality]] of the [[continuum]], $2^{\aleph_0}$.

Therefore these are all [[isomorphic]] as bare group. But equipped with their canonical Lie group structure (as in the [Examples](#Examples)) they are of course not isomorphic.

=--

### Different topologies on a Lie group

* Linus Kramer, _The topology of a simple Lie group is essentially unique_, ([arXiv](http://arxiv.org/abs/1009.5457))

> Abstract: We study locally compact group topologies on simple Lie groups. We show that the Lie group topology on such a group $S$ is very rigid: every 'abstract' isomorphism between $S$ and a locally compact and $\sigma$-compact group $\Gamma$ is automatically a homeomorphism, provided that $S$ is absolutely simple. If $S$ is complex, then non-continuous field automorphisms of the complex numbers have to be considered, but that is all.

### Which topological groups admit Lie group structure?

* _[[Hilbert's fifth problem]]_

### Relation to topological groups

* [[continuous homomorphisms of Lie groups are smooth]]

### Homotopy groups

List of [[homotopy groups]] of the manifolds underlying the classical [[Lie groups]] are for instance in ([Abanov 09](#Abanov09)).

## Applications

### In differential geometry

A central concept of [[differential geometry]] is that of a $G$-[[principal bundle]] $P \to X$ over a [[smooth manifold]] $X$ for $G$ a Lie group.

### In gauge theory

In the [[physics]] of [[gauge field]]s -- [[gauge theory]] -- Lie groups appear as local [[gauge group]]s parameterizing [[gauge transformation]]s: notably the [[Yang-Mills field]] is modeled by a $G$-[[principal bundle]] with [[connection on a bundle|connection]] for some Lie group $G$. For models that describe experimental observations the group $G$ in question is a quotient of a product of [[special unitary group]]s and the [[circle group]]. For details see [[standard model of particle physics]]

## In higher category theory

The notion of [[group]] generalizes in [[higher category theory]] to that of [[2-group]], ... [[∞-group]].

Accordingly, so does the notion of Lie group generalize to [[Lie 2-group]], ... [[∞-Lie group]]. For details see [[∞-Lie groupoid]].

## Examples
 {#Examples}

### Basic examples

* The [[real line]] $\mathbb{R}$ with its standard [[smooth structure]] and the group operation being addition is a Lie group. So is every [[Cartesian space]] $\mathbb{R}^n$ with the componentwise addition of real numbers.

* The [[quotient]] of $\mathbb{R}$ by the subgroup of [[integer]]s $\mathbb{Z} \hookrightarrow \mathbb{R}$ is the [[circle group]] $S^1 = \mathbb{R}/\mathbb{Z}$. The quotient $\mathbb{R}^n/\mathbb{Z}^n$ is the $n$-[[dimension|dimensional]] [[torus]].

* The [[automorphism group]] of any Lie group is canonically itself a Lie group: the _[[automorphism Lie group]]_.

### Classical Lie groups

The [[classical Lie groups]] include

* the [[general linear group]] $GL(n)$

* the [[orthogonal group]] $O(n)$ and [[special orthogonal group]] $SO(n)$;

* the [[unitary group]] $U(n)$ and [[special unitary group]] $SU(n)$;

* the [[symplectic group]] $Sp(2n)$.

### Exceptional Lie groups

The [[exceptional Lie groups]] incude

* [[G2]], [[F4]], [[E6]], [[E7]] [[E8]],

### Infinite-dimensional examples

* [[stable unitary group]]

* [[loop group]]

* [[diffeomorphism group]]

  * [[symplectomorphism group]], [[quantomorphism group]]

## Related concepts

* [[semisimple Lie group]], [[simple Lie group]], [[exceptional Lie group]]

* [[real form]]

* [[compact Lie group]], [[maximal compact subgroup]]

* [[maximal torus]]

* [[rank of a Lie group]]

* [[conjugacy class]], [[Cartan-Dirac structure on a Lie group]]

* [[Weyl group]]

* [[Inönü-Wigner group contraction]]

* [[infinite-dimensional Lie group]]

* [[orbit method]]

* [[Hamiltonian dynamics on Lie groups]]

* [[complex Lie group]]

* [[Poisson Lie group]]

* [[invariant differential form]], [[Maurer-Cartan form]]

[[!include infinitesimal and local - table]]

## References

### General

* {#Bredon72} [[Glen Bredon]], Section 0.5 of: _[[Introduction to compact transformation groups]]_, Academic Press  1972 ([ISBN 9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1), [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

  > (in the broader context of [[topological groups]])

* [[Nicolas Bourbaki]], _Lie groups and Lie algebras -- Chapters 1-3_, Springer (1975, 1989) [[ISBN:9783540642428](https://link.springer.com/book/9783540642428)]

* {#Adams82} [[Frank Adams]], *Lectures on Lie groups*, University of Chicago Press, 1982 ([ISBN:9780226005300](https://press.uchicago.edu/ucp/books/book/chicago/L/bo3614673.html), [gbooks](https://www.google.com/books/edition/Lectures_on_Lie_Groups/TC7d3ZcqjfsC?hl=en))

* {#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

* [[Tammo tom Dieck]], [[Theodor Bröcker]], Ch. I of: *Representations of compact Lie groups*, Springer 1985 ([doi:10.1007/978-3-662-12918-0](https://link.springer.com/book/10.1007/978-3-662-12918-0))

  > (in the context of [[representation theory]])

* [[José de Azcárraga]], [[José M. Izquierdo]], *[[Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics]]*, Cambridge Monographs of Mathematical Physics, Cambridge University Press (1995) [[doi:10.1017/CBO9780511599897](https://doi.org/10.1017/CBO9780511599897)]

* [[Howard Georgi]], *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

  > with an eye towards application to (the [[standard model of particle physics|standard model]] of) [[particle physics]]

* [[Hans Duistermaat]], [[Johan A. C. Kolk]], *Lie groups*, Springer (2000) [[doi:10.1007/978-3-642-56936-4](https://doi.org/10.1007/978-3-642-56936-4)]

* Mark Haiman, lecture notes by [[Theo Johnson-Freyd]], _Lie groups_, Berkeley 2009 ([pdf](http://math.berkeley.edu/~theojf/LieGroups.pdf))

* [[Eckhard Meinrenken]], _Lie groups and Lie algebas_, Lecture notes 2010 ([pdf](http://www.math.toronto.edu/mein/teaching/LectureNotes/lie.pdf))

* {#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))

* {#Hall15} [[Brian C. Hall]], *Lie Groups, Lie Algebras, and Representations*, Springer 2015 ([doi:10.1007/978-3-319-13467-3](https://doi.org/10.1007/978-3-319-13467-3))

* {#GallierQuaintance20a} [[Jean Gallier]], [[Jocelyn Quaintance]],  *Differential Geometry and Lie Groups -- A computational perspective*, Geometry and Computing **12**, Springer (2020) [[doi:10.1007/978-3-030-46040-2](https://doi.org/10.1007/978-3-030-46040-2), [webpage](https://www.cis.upenn.edu/~jean/gbooks/manif.html)]

* {#GallierQuaintance20b} [[Jean Gallier]], [[Jocelyn Quaintance]],  *Differential Geometry and Lie Groups -- A second course*, Geometry and Computing **13**, Springer (2020) [[doi:10.1007/978-3-030-46047-1](https://doi.org/10.1007/978-3-030-46047-1), [webpage](https://www.cis.upenn.edu/~jean/gbooks/manif.html)]

In the generality of [[Lie semigroups]]:

* [[Joachim Hilgert]], [[Karl-Hermann Neeb]], *Lie Semigroups and their Applications*. Lecture Notes in Mathematics **1552**, Springer 1993 ([doi:10.1007/BFb0084640](https://link.springer.com/book/10.1007/BFb0084640))

In the generality of [[quantum groups]]:

* [[Richard Borcherds]], [[Mark Haiman]], [[Theo Johnson-Freyd]], [[Nicolai Reshetikhin]], [[Vera Serganova]], *Berkeley Lectures on Lie Groups and Quantum Groups*, 2020 ([pdf](http://categorified.net/LieQuantumGroups.pdf))

### Homotopy groups

* Alexander Abanov, Homotopy groups of Lie groups 2009 ([pdf](http://felix.physics.sunysb.edu/~abanov/Teaching/Spring2009/Notes/abanov-cpA1-upload.pdf))
  {#Abanov09}

### On infinite-dimensional Lie groups
 {#ReferencesOnInfiniteDimensionalLieGroups}

References on [[infinite-dimensional Lie groups]]

* [[Andreas Kriegl]], [[Peter Michor]], _Regular infinite dimensional Lie groups_ Journal of Lie Theory
Volume 7 (1997) 61-99 ([pdf](http://www.heldermann-verlag.de/jlt/jlt07/MICHPL.PDF))

* Rudolf Schmid, _Infinite-Dimensional Lie Groups and Algebras in Mathematical Physics_ Advances in Mathematical Physics Volume 2010, ([pdf](http://www.emis.de/journals/HOA/AMP/Volume2010/280362.pdf))

* Josef Teichmann, _Infinite dimensional Lie Theory from the point of view of Functional Analysis_ ([pdf](http://www.math.ethz.ch/~jteichma/diss.pdf))

* [[Karl-Hermann Neeb]], _Monastir summer school: Infinite-dimensional Lie groups_ ([pdf](http://www.math.uni-hamburg.de/home/wockel/data/monastir.pdf))

### Spaces of homomorphisms

On [[mapping spaces]] of [[continuous function|continuous]] [[homomorphisms]] from [[topological groups]] to [[Lie groups]]:

for maps out of [[finitely generated group|finitely generated]] [[discrete groups]]":

* [[Frederick R. Cohen]], [[Mentor Stafa]], *A survey on spaces of homomorphisms to Lie groups*,  In: Callegaro F., Cohen F., De Concini C., Feichtner E., Gaiffi G., Salvetti M. (eds.) *Configuration Spaces*, INdAM Series, vol 14, Springer 2016 ([arXiv:1412.5668](https://arxiv.org/abs/1412.5668), [doi:10.1007/978-3-319-31580-5_15](https://doi.org/10.1007/978-3-319-31580-5_15))

for maps out of [[compact Lie groups]] and the fact that [[nearby homomorphisms from compact Lie groups are conjugate]]:

* [[Pierre Conner]], [[Edwin Floyd]], Ch. III, Lem. 38.1 in: _Differentiable periodic maps_, Ergebnisse der Mathematik und ihrer Grenzgebiete **33**, Springer 1964 ([doi:10.1007/978-3-662-41633-4](https://link.springer.com/book/10.1007/978-3-662-41633-4))

* [[Charles Rezk]], *Nearby homomorphisms from compact Lie groups are conjugate* ([MO:q/123624](https://mathoverflow.net/q/123624/381))

* {#Rezk14} [[Charles Rezk]], Rem. 2.2.1 in: *[[Global Homotopy Theory and Cohesion]]*, 2014 ([pdf](http://www.math.uiuc.edu/~rezk/global-cohesion.pdf), [[Rezk14.pdf:file]])

[[!redirects Lie groups]]

[[!redirects classical group]]
[[!redirects classical groups]]


Lie group cohomology

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
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#### Group Theory
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[[!include group theory - contents]]
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#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

_Lie group cohomology_ generalizes the notion of [[group cohomology]] from [[discrete group]]s to [[Lie group]]s.

From the [[nPOV]] on [[cohomology]], a natural definition is that for $G$ a Lie group, its cohomology is the intrinsic cohomology of its [[delooping]] [[Lie groupoid]] $\mathbf{B}G$ in the [[(∞,1)-topos]] $\mathbf{H} =$ [[smooth infinity-groupoid|Smth$\infty$Grpd]].

In the literature one finds a sequence of definitions that approach this intrisic topos-theoretic definition. This is discussed below. For a detailed discussion of the relation of this to the intrinsic topos-theoretic definition see the section Cohomology of Lie groups at [[∞-Lie groupoid]].

### Structured group cohomology

If the groups in question are not [[group object]]s internal to [[Set]] but groups with extra structure, such as [[topological group]]s or [[Lie group]]s, then their cohomology has to be understood in the corresponding natural context.

In parts of the literature cohomology of structured groups $G$ is defined in direct generalization of the formulas above as homotopy classes of morphisms from the simplicial object

$$
  \left(
    \cdots G \times G\stackrel{\to}{\stackrel{\to}{\to}}G \stackrel{\to}{\to} *
  \right)
$$

to a simplicial object $N (\mathbf{B}^n A)$.

This is what is described above. But this does **not** in general give the right answer for structured groups:

namely [[cohomology]] is really about homotopy classes of maps in the suitable ambient [[(∞,1)-topos]]. For plain groups as in the above entry, we are working in the $(\infty,1)$-topos [[∞Grpd]]. That may be modeled by the standard [[model structure on simplicial sets]]. In that model structure, all objects a cofibrant and [[Kan complex]]es are fibrant. That means all objects we are dealing with here are both cofibrant and fibrant, and hence the simplicial set of maps between them is the correct [[derived hom-space]] between these objects.

But this changes as we consider groups with extra structure. For a [[Lie group]] $G$, the object

$$
  \left(
    \cdots G \times G\stackrel{\to}{\stackrel{\to}{\to}}G \stackrel{\to}{\to} *
  \right)
$$

has to be considered as an [[Lie ∞-groupoid]]: an object in the [[model structure on simplicial presheaves]] over a [[site]] such as [[Diff]] or [[CartSp]]. As such it is in general **not** both cofibrant and fibrant. To that extent plain morphisms out of this object do **not** compute the correct [[derived hom-space]]s. Instead, the right definition of structured group cohomology uses the correct fibrant and cofibrant replacements.

### Topological group cohomology
 {#TopologicalGroupCohomology}

In

* [[Jim Stasheff]], _Continuous cohomology of groups and classifying spaces_  Bull. Amer. Math. Soc. Volume 84, Number 4 (1978), 513-530 ([web](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183540920))

$n$-cocycles on a topological group $G$ with valzues in a topological abelian group $A$ are considered as [[continuous functions]] $G^{\times n}\to A$ (p. 3 ). ("[[continuous cohomology]]")

A definition in terms of [[derived functor|Ext-functor]]s and comparison with the naive definition is in

* David Wigner, _Algebraic cohomology of topological groups_ Transactions of the American Mathematical Society, volume 178 (1973)([pdf](http://egg.epfl.ch/~nmonod/bonn/Wigner_1973.pdf))

A classical reference that considers the cohomology of Lie groups as topological spaces is

* [[Armand Borel]], _Homology and cohomology of compact connected Lie groups_ ([pdf](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063923/pdf/pnas01596-0040.pdf))

A corrected definition of topological group cohomology has been given by Segal

* [[Graeme Segal]], _Cohomology of topological groups_ In Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), pages 377{387. Academic Press, London, (1970).

* [[Graeme Segal]], _A classifying space of a topological group in the sense of Gel'fand-Fuks. Funkcional. Anal. i Prilozen., 9(2):48{50, (1975).

## Definition

For $G$ a [[Lie group]] and $A$ an abelian Lie group, write

$$
  H_{naive}^n(G,A) = \{smooth G^{\times n } \to A\}/_\sim
$$

for the naive notion of cohomology on $G$.

A refined definition of Lie group cohomology, denoted $H^n_{diff}(G,A)$, was given in ([Brylinski](#Brylinski)) following ([Blanc](#Blanc)) and effectively rediscovers Segal's definition. See section 4 of ([Schommer-Pries](#Shommer-Pries)) for a review and applications.

+-- {: .un_defn}
###### Definition
**(Brylinski)**

Let $G$ be a [[Lie group]] ([[paracompact space|paracompact]]) and $A$ an abelian Lie group.

For eack $k \in \mathbb{N}$ we can pick a [[good open cover]] $\{U^{k}_{i} \to G^{\times_k}| i \in I_k\}$ such that

* the index sets arrange themselves into a [[simplicial set]]
  $I : [k] \mapsto I_k$;

* and for $d_j(U^k_i)$ and $s_j(U^k_i)$ the images of the
  face and degeneracy maps of $G^{\times\bullet}$ we have

  $$
    d_j(U^k_i) \subset U^{k-1}_{d_j(i)}
  $$

  and

  $$
    s_j(U^k_i) \subset U^{k+1}_{s_j(i)}
    \,.
  $$

Then the **differentiable group cohomology** of $G$ with coefficients in
$A$ is the cohomology of the total complex of the [[Cech cohomology|Cech]]
double complex $C^\infty( U^{\bullet}_{i_0, \cdots, i_\bullet} , A)$
whose differentials are the alternating sums of the face maps
of $G^{\times_\bullet}$ and of the [[Cech nerve]]s, respectively:

$$
  H^n_{diff}(G,A) := H^n Tot C^\infty( U^{\bullet}_{i_0, \cdots, i_\bullet} , A)
$$

=--

This is definition 1.1 in ([Brylinski](#Brylinski))

As discussed there, this is equivalent to other definitions, notably to a definition given earlier by [[Graeme Segal]].

There is an evident morphism

$$
  H^n_{naive}(G,A) \to H^n_{diff}(G,A)
$$

obtained by pulling back a globally defined smooth cocycle to a cover.

At [[∞-Lie groupoid]] it is discussed that there is a further refinement

$$
  H^n_{diff}(G,A) \to H^n(\mathbf{B}G,A)
  \,,
$$

where on the right we have the [[cohomology|intrinsic cohomology]] of [[∞-Lie groupoid]]s.

## Properties

### General

From now on, for definiteness by a _Lie group_ $G$ we mean (following [Bry, page3](http://arxiv.org/PS_cache/math/pdf/0011/0011069v1.pdf#page=3)) a [[paracompact space|paracompact]] [[Frechet manifold]] equipped with a group structure such that the product and the inverse maps are smooth, and there is an everywhere defined exponential map $exp : \mathfrak{g} \to G$ where $\mathfrak{g}$ is the [[Lie algebra]] of $G$.

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

If the coefficient Lie group $A$ is a [[topological vector space]], then the naive group cohomology $H^n(G,A) = \{smooth G^{\times n} \to A\}/_\sim$ coincides with the correct Lie group cohomology

$$
  (A top.\;vect.\;space) \Rightarrow
  (H^n_{naive}(G,A) \stackrel{\simeq}{\to} H^n_{diff}(G,A) )
  \,.
$$

If the coefficient Lie group $A$ is discrete, then Lie group cohomology coindices with the topological cohomology of the [[classifying space]] $\mathcal{B}G$

$$
  (A discrete) \Rightarrow
  (H^n(G,A) \simeq H^n(\mathcal{B}G), A)
  \,.
$$

=--

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

This is [Bry, prop. 1.3](http://arxiv.org/PS_cache/math/pdf/0011/0011069v1.pdf#page=4) and [Bry, lemma 1.5](http://arxiv.org/PS_cache/math/pdf/0011/0011069v1.pdf#page=5).

=--

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

$H^2_{diff}(G,A)$ classifies central [[group extension|extensions of Lie groups]]

$$
   A \to \hat G \stackrel{\pi}{\to} G
$$

such that $\pi : \hat G \to G$ is a locally trivial smooth [[principal bundle|principal]] $A$-fibration.

The image of $H^2_{naive}(G,A) \to H^2_{diff}(G,A)$ consists of those central extensions for which is bundle is trivial.

=--

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

This is [Bry, prop. 1.6](http://arxiv.org/PS_cache/math/pdf/0011/0011069v1.pdf#page=4) and [Bry, lemma 1.5](http://arxiv.org/PS_cache/math/pdf/0011/0011069v1.pdf#page=5).

=--

### Long sequence in differential cohomology {#SequenceDiffCohomology}

For the purpose of this section we specifically conceive Lie group cohomology inside the [[(∞,1)-topos]] [[∞LieGrpd]] of [[∞-Lie groupoid]]s, as described there.

This is a [[local (∞,1)-topos]], hence in particular an [[∞-connected (∞,1)-topos]] and therefore it admits [[schreiber:differential cohomology in an (∞,1)-topos]]. By the theorem about the differential fiber sequence
we have for $G$ a [[Lie group]], $\mathbf{B}G$ its [[delooping]], $\mathbf{B}^{n} U(1)$ the [[∞-Lie groupoid|circle n+1-group]] a long sequence in cohomology

$$
  \cdots \to H^n_{diff}(G, (1))\to H^n_G(G,U(1)) \to H_{dR, G}^{n+1}(G) \to
  \cdots
  \,,
$$

where $H_G(G,-)$ denotes $G$-equivariant cohomology, in that

$$
  H^n_G(G,A) := \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^n A)
$$

and so on.

The point to note is that we may identify [[Lie algebra cohomology]] inside $H^n_{dR,G}(G)$ and may therefore regard the map

$$
  H^n_G(G,U(1)) \to H_{dR, G}^{n+1}(G)
$$

as the differentiation map that take a smooth group cocycle to a Lie algebra cocycle. This morphism operates by putting constructing a [[circle n-bundle with connection]] over $\mathbf{B}G$ and then computing its [[curvature]] forms.

**Example**

For $\mathfrak{g}$ a [[semisimple Lie algebra]],$G$ its simply connected Lie group,  let $\mathbf{B}G \to \mathbf{B}^3 U(1)$ be the group cocycle that classifies the [[string 2-group]]. Its image in $H^3_{dR,G}(G)$ is the curvature of the [[Chern-Simons circle 3-bundle]] over $\mathbf{B}G$. This is represented by a simplicial differential form consisting of two pieces

* on $G$ the form $\langle \theta\wedge \theta \wedge \theta \rangle$ obtained by feeding the [[Maurer-Cartan form]] on $G$ into the canoical Lie algebra cocycle that is [[Chern-Simons element|in transtression]] with the [[Killing form]] [[invariant polynomial]];

* on $G \times G$ something like $\langle \theta_1 \wedge \theta_2\rangle$.

(...)

### Relation to intrinsic cohomology of smooth $\infty$-groupoids

We may naturally regard a Lie group as an [[∞-group]] in the
[[cohesive (∞,1)-topos]] [[Smooth∞Grpd]] of [[smooth ∞-groupoid]]s. As such, there is an intrinsic [[(∞,1)-topos]]-theoretic notion of its [[cohomology]].

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

For

1. $G$ a [[Lie group]] and $A$ either a [[discrete group]]

1. $G$ a [[compact topological space|compact]] [[Lie group]] and $A$ the additive Lie group of [[real numbers]] $\mathbb{R}$ or the [[circle group]] $\mathbb{R}/Z = U(1)$

the intrinsic cohomology of $G$ in [[Smooth∞Grpd]] coincides with the refined [[Lie group cohomology]] of ([Segal](#Segal))([Brylinski](#Brylinski))

$$
  H^n_{Smooth\infty Grpd}(\mathbf{B}G, A) \simeq H^n_{diffr}(G,A)
  \,.
$$

=--

This is discussed in detail at [[Smooth∞Grpd]] and proven at [[SynthDiff∞Grpd]].

### Relation to Lie algebra cohomology

The content of a [[van Est isomorphism]] is that the canonical comparison map from Lie group cohomology to [[Lie algebra cohomology]] (by [[differentiation]]) is an [[isomorphism]] whenever the Lie group is sufficiently connected.

## Related entries

* [[group cohomology]]

  * [[nonabelian group cohomology]], [[groupoid cohomology]]

  * [[∞-group cohomology]]

* [[group extension]]

* **Lie group cohomology**

  * [[∞-Lie groupoid cohomology]]

## References

A textbook account of standard material is in chapter V in vol III of

* [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], _[[Connections, Curvature, and Cohomology]]_ Academic Press (1973)

The definition of the refined [[topological group]] cohomology in terms of degreewise [[abelian sheaf cohomology]] was given in

* [[Graeme Segal]], _Cohomology of topological groups_ , Symposia Mathematica, Vol IV (1970) (1986?) p. 377

It was later rediscovered for [[Lie group]]s in

* {#Brylinski} [[Jean-Luc Brylinski]], _Differentiable Cohomology of Gauge Groups_ ([arXiv](http://arxiv.org/abs/math/0011069))

following

* {#Blanc}P. Blanc, _Cohomologie différentiable et changement de groupes, Astérisque vol. 124-125 (1985), pp. 113-130

Further discussion:

* [[Friedrich Wagemann]], [[Christoph Wockel]], *A Cocycle Model for Topological and Lie Group Cohomology*, Transactions of the American Mathematical Society **367** 3 (2015) 1871-1909 ([arXiv:1110.3304](https://arxiv.org/abs/1110.3304), [jstor:24513021](https://www.jstor.org/stable/24513021))

Relevant background on the theory of abelian sheaf cohomology on simplicial spaces is at the beginning of

* [[Pierre Deligne]], _Théorie de Hodge: III_  Publication mathétematique de l'I.H.E.S, tome 44 (1974), p. 5-77 ([numdam](http://www.numdam.org/item?id=PMIHES_1974__44__5_0))

More discussion of Lie group cohomology along these lines is in

* [[Chris Schommer-Pries]], _A finite-dimensional String 2-group_ ([arXiv:0911.2483](http://arxiv.org/abs/0911.2483))
{#Schommer-Pries}

A discussion of the relation between _local_ Lie group cohomology and [[Lie algebra cohomology]] is in

* S. Świerczkowski, _Cohomology of group germs and Lie algebras_ Pacific Journal of Mathematics, Volume 39, Number 2 (1971), 471-482.  ([pdf](http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pjm/1102969572))

[[!redirects differentiable Lie group cohomology]]


Lie groupoid

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#Contents#
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## Idea

A _Lie groupoid_ is a [[groupoid]] [[internal groupoid|internal]] to [[smooth manifolds]]. This is a joint generalization of [[smooth manifolds]] and [[Lie groups]] to [[higher differential geometry]].

Regarded in the more general context of [[smooth groupoids]]/[[smooth stacks]], Lie groupoids present certain well-behaved such objects, often called _[[differentiable stacks]]_.

## Definition

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

A _Lie groupoid_ $X \coloneq (X_1 \rightrightarrows X_0)$ is a [[groupoid]] such that both the space of arrows $X_1$ and the space of objects $X_0$ are smooth manifolds, all structure maps are smooth,  and source and target maps $s, t: X_1\rightrightarrows X_0$ are surjective submersions.

=--

A _Lie groupoid_ $X$ is an [[internal groupoid]] in the [[category]] [[Diff]] of [[smooth manifolds]].

Since [[Diff]] does not have all [[pullbacks]], to ensure that this definition makes sense, one needs to ensure that the space $X_1 \times_{s,t} X_1$ of composable [[morphism]]s is an object of [[Diff]]. This is achieved either by adopting the definition of [[internal groupoid]] in the sense of Ehresmann, which includes as data the [[smooth manifold]] of [[composable pairs]], or by taking the conventional route and demanding that the source and target maps $s,t : X_0 \to X_1$ are [[submersion]]s. This ensures the [[pullback]] exists to define said manifold of composable pairs. Therefore a definition used in most modern differential geometry literature is as we see above.

But for most practical purposes, the apparently evident [[2-category]] $Grpd(Diff)$ of such internal groupoids, [[internal functor]]s and internal [[natural transformation]]s is _not_ the correct 2-category to consider. One way to see this is that the [[axiom of choice]] fails in [[Diff]], which means that an internal functor $X \to Y$ of internal groupoids which is [[essentially surjective functor|essentially surjective]] and [[full and faithful functor|full and faithful]] may nevertheless not be an equivalence, in that it may not have a weak inverse in $Grpd(Diff)$.

See the section [2-Category of Lie groupoids](#2CatOfGrpds) below.

A bit more general than a Lie groupoid is a [[diffeological groupoid]].

### Examples for Lie groupoids

* A Lie group $G$ is a Lie groupoid with $X_1=G$ and $X_0=pt$ a point. Composition of $X$ is provided by the multiplication of $G$.

* A manifold $M$ is a Lie groupoid with $X_1=M$ and $X_0=M$. Source and target maps are identities and we only have identity arrows in this example.

* Given a manifold $M$ and an open cover $\{U_i\}$, we can form a Lie groupoid with $X_1=\sqcup U_i\times_M U_j$ and $X_0=\sqcup U_i$. Then for an element $x_{ij}:=(x_i, x_j)\in U_i\times_M U_j \subset X_1$, $t(x_{ij})=x_i \in U_i$, $s(x_{ij})=x_j \in U_j$, and $x_{ij} \cdot x_{jk}= x_{ik}$. This is sometimes called the [[Čech groupoid]] or **covering groupoid**.

* Given a Lie group $G$ (right) action  on a manifold $M$, then we may form an associated [[action groupoid]] (or sometimes called **transformation groupoid**) as follows: $X_1 = M \times G$ and $X_0=M$. For an element $(x, g) \in X_1$, we have $t(x, g) = x$, $s(x, g)=x\cdot g^{-1}$,  and $(x, g)\cdot (y, h) = (x, g\cdot h)$ (we must have $y=x\cdot g^{-1}$ for the multiplication to happen). Action groupoid presents the quotient stack $[M/G]$. Roughly speaking, it is a good replacement for quotient space even if the action is not as nice as you want.

* Given a manifold $M$, we may also form so-called [[pair groupoid]]: $X_1= M\times M$ and $X_0=M$. Source and target are projections, and multiplication is given by $(x, y) \cdot (y , z)= (x, z)$. Pair groupoid may be interpreted as the global object of tangent bundle (think why? see the section below on Lie algebroid).

* Given a manifold $M$, we have also an associated [[fundamental groupoid]] or **homotopy groupoid** $\Pi(M)$: $\Pi(M)_1=\{$paths in $M\}/$ homotopies, $\Pi_0(M)=M$. Source and target are end points of a path. Multiplication is concatenation of paths (think why associative?).

* Given a manifold $M$ with a [[foliation]] $F$, we may form various groupoids associated with $F$.

1. **$F$-pair groupoid**: $X_1:=\{(x, y)| x, y \;\text{are in the same leaf in}\; F \}$, $X_0=M$. Source and target are obvious projections and multiplication is like in the case of pair groupoid. The problem for this groupoid is that it might not be  a Lie groupoid. (why not? for counter example, we refer to [Section 13.5 of Geometric Models for
Noncommutative Algebras](https://math.berkeley.edu/~alanw/Models.pdf) ).

1. **monodromy groupoid $Mon_F(M)$** (it is a foliation version of fundamental groupoid, thus it is also sometimes called $F$-fundamental groupoid): $X_1:=\{$ leaf-wise paths$\}/$ leaf-wise homotopy, $X_0=M$ and the rest is like in the case of fundamental groupoid.

1. **[[holonomy groupoid]] $Hol_F(M)$**: $X_1:=\{$ leaf-wise paths$\}/$ holonomy,
$X_0=M$ and the rest is like in the case of fundamental groupoid. Here, the holonomy of a path $\gamma$ is defined to the germ of diffeomorphisms induced by $\gamma$ between the transversals at the end points.

Among all possible Lie groupoids associated to a foliation, monodromy groupoid is the biggest and holonomy groupoid is the smallest.

### Terminology

Originally Lie groupoids were called (by Ehresmann) _differentiable groupoids_ (and also one considered differentiable _categories_). Sometime in the 1980s there was a change of terminology to _Lie groupoid_ and [[differentiable stack]]s. (reference?)

### Specialisations

One definition which Ehresmann introduced in his paper _Catégories topologiques et catégories différentiables_ (see below) is that of [[locally trivial category|locally trivial groupoid]]. It is defined more generally for topological categories, and extends in an obvious way to topological groupoids, and Lie categories and groupoids. For a topological (resp. Lie) category $X$, let $X_1^{iso}$ denote the subspace (resp. submanifold) of invertible arrows . (This always exists, by general abstract nonsense - I should look up the reference, it's in Bunge-Pare I think - [[David Roberts|DR]])

+-- {: .num_defn}
###### Definition
A topological groupoid $X_1 \rightrightarrows X_0$ is **locally trivial** if for every point $p\in X_0$ there is a neighbourhood $U$ of $p$ and a lift of the inclusion $\{p\} \times U \hookrightarrow X_0 \times X_0$ through $(s,t):X_1^{iso}\to X_0 \times X_0$.
=--

Clearly for a Lie groupoid $X_1^{iso} = X_1$. It is simple to show from the definition that for a transitive Lie groupoid, $(s,t)$ has local sections. Ehresmann goes on to show a link between smooth [[principal bundles]] and transitive, locally trivial Lie groupoids. See [[locally trivial category]] for details.

### The (2,1)-category of Lie groupoids {#2CatOfGrpds}

As usual for internal categories, the naive 2-category of internal groupoids, [[internal functor]]s and internal [[natural transformation]]s is not quite "correct". One sign of this is that the [[axiom of choice]] fails in [[Diff]] so that an internal functor which is an [[essentially surjective functor]] and a [[full and faithful functor]] may still not have an internal weak inverse.

One way to deal with this is to equip the 2-category with some structure of a [[homotopical category]] and allow morphisms of Lie groupoids to be [[anafunctor]]s, i.e. [[span]]s of internal functors $X \stackrel{\simeq}{\leftarrow} \hat X \to Y$.

Such generalized morphisms -- called _[[Morita morphisms]]_ or _generalized morphisms_ in the literature -- are sometimes modeled as [[bibundle]]s and then called [[Hilsum-Skandalis morphism]]s.

Another equivalent approach is to embed Lie groupoids into the context of [[2-topos]] theory:

The [[2-topos|(2,1)-topos]] $Sh_{(2,1)}(Diff)$ of [[stack]]s/[[2-sheaves]] on [[Diff]] may be understood as a nice [[2-category]] of general groupoids _modeled on_ [[smooth manifold]]s. The degreewise [[Yoneda embedding]] allows to emebed groupoids internal to $Diff$ into stacks on $Diff$. this wider context contains for instance also [[diffeological groupoid]]s.

Regarded inside this wider context, Lie groupoids are identified with [[differentiable stack]]s. The [[(n,r)-category|(2,1)-category]] of Lie groupoids is then the full sub-$(2,1)$-category of $Sh_{(2,1)}(Diff)$ on differentiable stacks.

For more comments on this, see also the beginning of [[∞-Lie groupoid]].

## Lie algebroid

As the [[infinitesimal space|infinitesimal]] approximation to a [[Lie group]] is a [[Lie algebra]], so the infinitesimal approximation to a Lie groupoid is a [[Lie algebroid]].

+-- {: .num_defn}
###### Definition
A Lie algebroid is a vector bundle $A\to M$ together with a vector bundle morphism $\rho: A\to TM$ (called anchor map), and a Lie bracket $[-,-]$ on the space of sections of $A$, satisfying the Leibniz rule

$[X, fY]=f[X,Y]+\rho(X)(f) Y. $
=--
+-- {: .num_defn}
###### Remark
You would expect  $\rho$ to preserve $[-,-]$, wouldn't you? It is actually automatic! (see Y. Kosmann-Shwarzbah and F. Magri. Poisson-Nijenhuis strutures. Ann. Inst.
H. Poinaré Phys. Théor., 53(1):3581, 1990.)

Recent progress: it turns out that one may link Lie algebroid with $L_\infty$-spaces (ask Owen Gwilliam for it)
=--

### Examples of Lie algebroids

* A Lie algebra is a Lie algebroid with base space being a point.

* $0$-bundle over a manifold $M$ is certainly a Lie algebroid in a trivial way.

* [[action Lie algebroid]]

* Tangent bundle $TM\to M$ is a Lie algebroid with $\rho=id$ and $[-,-]$ the usual Lie bracket for vector fields. See [[tangent Lie algebroid]].

* Given a [[Poisson manifold]] $P$ with Poisson bivector field $\pi$, the cotangent bundle $T^*P$ is equipped with a Lie algebroid structure: $\rho(\xi)=
\pi(\xi)$ and $[\xi_1, \xi_2]=d\pi(\xi_1, \xi_2)$ (or you may have $[df, dg]=d\{ f, g\}$ if you prefer to think in Poisson bracket). See [[Poisson Lie algebroid]].

* [[Atiyah Lie algebroid]]

## Morphisms of Lie groupoids
 {#MorphismsOfLieGroupoids}

There are several versions of Lie groupoid morphisms, some of them are equivalent in a correct sense, some of them are not.

* strict morphism: a strict morphism from Lie groupoid $X$ to $Y$ is a functor from $X$ to $Y$ as categories and preserving the smooth structures.

* generalised morphism: a generalised morphism from $X$ to $Y$ is a [[span]] of morphisms $X \stackrel{\simeq}{\leftarrow} \hat X \to Y$, where $\hat X \stackrel{\simeq}{\rightarrow} X$ is a [[weak equivalence]] of Lie groupoids, defined as below (see also at _[[bibundle]]_).

A Lie groupoid functor $f : G\to H$ is a  **weak equivalence** if it is

1. essentially surjective; that is, $t \circ pr_2 : G_0 \times_{H_0,s} H_1 \to H_0$ is a surjective submersion;

1. fully faithful; that is,  $G_1 \cong H_1\times_{t\times s, H_0\times H_0} G_0 \times G_0$.

Composition of generalised morphism is given by weak [[pullback]] of Lie groupoids (see also [[weak limit]]). Given  (strict) morphisms $\hat X\to Y$ and $\hat X' \to Y$, the **weak [[pullback]]** of $\hat X\to Y$ along $\hat X' \to Y$ is a groupoid $\hat X \times_{Y}^w \hat X'$ with space of objects $\hat X_0 \times_{ Y_0} Y_1 \times_{Y_0} \hat X'_0$ and space of morphisms $\hat X_1 \times_{Y_0} Y_1 \times_{Y_0} \hat X'_1$. When $\hat X' \to Y$ is a weak equivalence, the weak pullback is a Lie groupoid thank to the property of essentially surjective. (Is this composition associative?)

* [[anafunctor]]: an anafunctor from $X$ to $Y$ is a [[span]] of morphisms $X \stackrel{\simeq}{\leftarrow} \hat X \to Y$, where $\hat X \stackrel{\simeq}{\rightarrow} X$ is an [[acyclic fibration]] of Lie groupoids. That is, this map is a [[weak equivalence]] of Lie groupoids and $\hat X_0 \to X_0$ is a surjective submersion.

Composition of anafunctors is given through strong [[pullback]] of Lie groupoids, that is level-wise pullback.

* [[bibundle]] functor (or H.S. bibundle, or Hilsum-Skandalis bibundle): a bibundle functor from $G\to H$ is a [[groupoid principal bundle]]  $E$ of $H$ (with right action) such that $G$ acts on $E$ from left and $G$ action commutes with $H$ action. If both $G$ and $H$ actions are principal, then $E$ gives arise to [[Morita equivalence]] between them.

The last three morphisms are more or less equivalent, that is they give arise to equivalent 2-categories (in fact (2,1)-categories) of Lie groupoids. To make it explicit, we need to talk about [[2-morphism]]s between them.

A [[2-morphism]] between bibundle functors is simply a bibundle isomorphism (of course preserving all the structures of bibundles).

A strict [[2-morphism]] from generalised morphism $X \stackrel{\simeq}{\leftarrow} \hat X \to Y$ to $X \stackrel{\simeq}{\leftarrow} \hat X' \to Y$ is given by a morphism $\hat X \to \hat X'$ such that  the following diagram commutes
$$
\begin{matrix}
  &          & \hat X \\
  & \swarrow &           & \searrow \\
X &          & \downarrow &          & Y \\
  & \searrow &           & \swarrow \\
  &          & \hat X'
\end{matrix}
$$
This forces the morphism $\hat X \to \hat X'$ to be a weak equivalence by [[2-out-of-3]] for weak equivalences. A [[2-morphism]] from $X \stackrel{\simeq}{\leftarrow} \hat X \to Y$ to $X \stackrel{\simeq}{\leftarrow} \hat X' \to Y$ is provided by a [[span]] of strict 2-morphisms:

$$
\begin{matrix}
  &          & \hat X \\
  & \swarrow &     \uparrow      & \searrow \\
X &          & \hat X'' &          & Y \\
  & \searrow &       \downarrow    & \swarrow \\
  &          & \hat X'
\end{matrix}
$$

A [[2-morphism]] between [[anafunctor]]s are defined like above, however the left legs are required to be [[acyclic fibration]]s between Lie groupoids.  (think this time what may you say about the morphism $\hat X \to \hat X'$?)

Then these three (2,1)-categories, which we denote by $GEN$, $ANA$ and $BUN$, are  all **[[equivalent]]** to each other. For a nice survey on this statement, we refer to Section 1.5 of [Du Li's thesis](http://ediss.uni-goettingen.de/handle/11858/00-1735-0000-0022-5F4F-A).

The idea is that  [Bundlisation](http://ncatlab.org/nlab/show/bibundle#bundlisation) may extend to an equivalence of $(2,1)$-categories between $GEN$, the $(2,1)$-category made by generalised morphisms,  and $BUN$. The inverse is given by the following construction: given a bibundle functor $E: G\to H$, we pull back $G$ along the map $E\to G_0$ and obtain a Lie groupoid $G|_E:=G_1\times_{G_0\times G_0} E \times E \Rightarrow E$. Then the natural projection  $G|_E \to G$ is an [[acyclic fibration]]. Thus we obtain a generalised morphism which is also an anafunctor from $G \to H$.

Even though $GEN$ contains more morphisms than $ANA$, a generalised morphism maybe equivalently replaced by an anafunctor. In fact a generalised morphism $X \stackrel{\simeq}{\leftarrow} \hat X \to Y$  gives arise to an anafunctor $X \stackrel{\simeq}{\leftarrow} X \times_{X}^w \hat X \to Y$.

As a consequence of the universal property of the [[calculus of fractions]], $GEN$ and $ANA$ are equivalent.

## Morphisms of Lie algebroids

Morphisms of Lie algebroids are counter-intuitive: they are not morphisms of vector bundles which preserve the algebroid structure. To define a Lie algebroid morphism, we first need to introduce the _Chevalley-Eilenberg algebra_ associated to a Lie algebroid $A$.

We consider $A$ to be a trivially graded vector bundle, i.e. concentrated in degree $0$. Then $A[1]$ is concentrated in degree $-1$.
The functions on $A[1]$ are given as
$$C(A[1])=C^\infty(M)\oplus \Gamma(A^*)\oplus \Gamma(\wedge^2 A^*)\oplus \ldots ,$$
where $C^\infty(M)$ is considered to be of degree $0$, $\Gamma(A^*)$ to be of degree $1$, and so forth.

Now we can define a degree-one derivation on $C(A[1])$ as follows: For $\xi \in \Gamma(\wedge^n A^*)$ and $X_i\in \Gamma(A)$, let
$$
d_A(\xi)(X_1,\,\ldots\,,\,X_n) := \sum_{0\leq i \lt j\leq n} (-1)^{i+j} \xi\bigl([X_i,\,X_j]_A,\, \ldots\,,\, \widehat{X_i},\,\ldots\,,\,\widehat{X_i},\,\ldots\bigr) + \sum_{i=0}^n (-1)^i \rho_A(X_i) \xi\bigl(\ldots\,,\,\widehat{X_i},\,\ldots\bigr).
$$
The condition $[d_A,\,d_A] = 0$ is not automatically fulfilled: since $\deg d_A = 1$, we have $[d_A,\,d_A] = d_A \circ d_A + d_A \circ d_A = 2 d_A \circ d_A$.
The condition $d_A \circ d_A = 0$ is actually equivalent to $\bigl(A, \rho_A, [ - , - ]_A\bigr)$ being a Lie algebroid; that is, it is fulfilled if and only if

 * $[- , - ]_A$ satisfies the Jacobi identity
 * and $[ - , - ]_A$ and $\rho_A$ together satisfy the Leibniz identity.

(Proof: calculation gives the restriction $d_A(f) = \rho^\ast(d f)$
and $d_A(\xi)(X_1, X_2)
 = - \bigl\langle \xi, [X_1, X_2]\bigr\rangle
   + \rho_A(X_1)(\xi X_2) - \rho_A(X_2)(\xi X_1)$
and $d_A(\xi)(X_1, X_2, X_3)
 = - \xi([X_1, X_2], X_3) + \xi([X_1, X_3], X_2) - \xi([X_2, X_3], X_2)
   + \rho_A(X_1)\xi(X_2, X_3) - \rho_A(X_2)\xi(X_1, X_3) + \rho_A(X_3)\xi(X_1, X_2)$.)

This point of view also applies to [[higher Courant Lie algebroids]] and [[L-infinity-algebra]]s.

Example: For the tangent Lie algebroid $A = T M$, $\bigl(C(T M[1]), d_A\bigr) = \bigl(\Omega^\ast(M), d_{dR}\bigr)$.

Then a morphism from a Lie algebroid $(A, \rho_A, [-,-]_A)$ to $(B, \rho_B, [-,-]_B)$ is a morphism of the associated differential graded commutative algebras

$$ (C(A[1]), d_A) \leftarrow (C(B[1]), d_B).
$$

Such a morphism of c.d.g.a.'s is determined by maps $C^\infty(N) \to C^\infty (M)$ on degree $0$ and a map $\Gamma(B^*)\to \Gamma(A^*)$ on degree $1$. Thus a morphism of vector bundles $A\xrightarrow{f} B$ give rise to a morphism $f^*$ of c.g.a. For $f$ to be a Lie algebroid morphism, we further need $f$ to  satisfy additional conditions so that $f^*$ preserves the differential.

This way to explain morphisms of Lie algebroids is described in [[Kirill Mackenzie]], chapter 4.3.
If the Lie algebroids are over the same manifold $M$, then a morphism from $A$ to $B$ can be described as a morphism of vector bundles that respects the anchor maps and the Lie bracket. If, however, $B$ is over a different manifold $N$, this direct approach does not work. In this situation we have to pull back the Lie algebroid to $M$ (Note that this is not simply the vector bundle pullback of $B$ along $f$, but a more involved construction, see [[Kirill Mackenzie]]). Using the defintion of a morphism on a common base manifold one arrives at two conditions on the bundle morphism to be a morphism of Lie algebroids. For details see the linked book.

### Example
Let $I$ be an interval with the tangent bundle Lie algebroid $(TI, \id_{TI}, [-,-])$ and $(A, \rho_A, [-,-]_A)$ an arbitrary Lie algebroid on $M$. Then a path $a\colon I \to A$ defines a map from $\varphi\colon C(A[1]) \to C(TI[1]) \cong \Omega(I)$ which respects the commutative graded algebra structure. A function $f\in C^{\infty}(M)$ gets mapped to $f\circ \gamma$, where $\gamma\colon I \to M$ is the projection of $a$. A section $s\in\Gamma(A^*)$ gets mapped to $(s\circ \gamma (a) dt$ under $\varphi$, where $dt$ is the canonical section of $\Omega^1(I)$. Since $C(TI[1])$ is concentrated in degree $0$ and $1$, the other degrees get mapped to $0$.

For this map to be a morphism of Lie algebroids, it has to respect the differentials. As explained above this only needs to be checked for a smooth function in $C^{\infty}(M)$:

$$ \varphi(d_A f) = d_{dR} \varphi(f) \qquad f\in C^{\infty}(M).
$$

A quick calculation shows that this is true if and only if

$$ \rho_A(a(t)) = \frac{d}{dt}\gamma(t).
$$

### Example
Let $M$ the unit square in $\mathbb{R}^2$. Then a Lie algebroid morphism from $(TM, \id_{TM}, [-,-])$ to $(\mathfrak{g}, T\cdot, [-.-]_{\mathfrak{g}})$, where $\mathfrak{g}$ is a Lie algebra, is given by a morphism of c.d.g.a.

$$ (CE(\mathfrak{g}), d_{\mathfrak{g}}) \to (\Omega(M), d_{dR}).
$$

Two smooth maps $a,b\colon M\to \mathfrak{g}$ give a map between c.g.a. spaces above. Here a section $s\in\Gamma(\mathfrak{g}^*)$, i.e. an element of $\mathfrak{g}^*$ gets mapped to the one form

$$ s(a) dt + s(b) ds,
$$

where $(t,s)$ are the coordinates on $M$. A quick calculation shows that this map respects the differential if and only if

$$ \frac{da}{ds} - \frac{db}{ds} = [a,b]_{\mathfrak{g}}.
$$

This formula also shows that $a\cdot dt + b\cdot ds$ is a flat connection on the trivial principal $G$-bundle on $M$. Therefore Lie algebroid morphisms open a way to talk about higher connections and flat conditions.

### Open problem
It is unclear how to use the idea of [[bibundle]] or [[span]] to define a more general version of Lie algebroid morphisms so that they really correspond to the case of Lie groupoid morphisms.

## Higher Lie groupoids

See

* [[internal ∞-groupoid]]

* [[∞-Lie groupoid]]

## Examples

* Every [[smooth manifold]] $X$ is a 0-[[0-groupoid|truncated]] Lie groupoid.

* For every [[Lie group]] $G$ the one-object [[delooping]] groupoid $\mathbf{B}G$ is a Lie groupoid.

* The Lie group $G$ itself is a 0-[[truncated]] [[group object]] in the 2-category or Lie groupoids.

* Every [[Lie 2-group]] is in particular a Lie groupoid: a [[group object]] in the category of Lie groupoids.

* The [[inner automorphism 2-group]] $\mathbf{E}G = INN(G) = G//G$ is a Lie groupoid. There is an obvious morphism $\mathbf{E}G \to \mathbf{B}G$.

* For every $G$-[[principal bundle]] $P \to X$ there is its [[Atiyah Lie groupoid]] $At(P)$.

* The [[fundamental groupoid]] $\Pi_1(X)$ of a smooth manifold is naturally a Lie groupoid.

* The [[path groupoid]] of a smooth manifold is naturally a [[diffeological groupoid]].

* The [[Cech groupoid]] $C(U)$ of a [[cover]] $\{U_i \to X\}$ of a smooth manifold is a Lie groupoid.

* Every [[foliation]] gives rise to its [[holonomy groupoid]].

* An [[orbifold]] is a Lie groupoid.

* An [[anafunctor]] $X \stackrel{\simeq}{\leftarrow} C(U) \to \mathbf{B}G$ from a smooth manifold $X$ to $\mathbf{B}G$ is a [[Cech cohomology|Cech cocycle]] in degree 1 with values in $G$, classifying $G$-[[principal bundle]] $P$.

* The (1-categorical) [[pullback]]

  $$
    \array{
       P &\stackrel{\simeq}{\leftarrow}& \mathbf{P} &\to& \mathbf{E}G
       \\
       && \downarrow && \downarrow
       \\
       && C(U) &\stackrel{}{\to}& \mathbf{B}G
       \\
       && \downarrow^{\mathrlap{\simeq}}
       \\
       && X
    }
  $$

  is a Lie groupoid equivalent to this principal bundle $P$.

  (For more on the general phenomenon of which this is a special case see [[principal ∞-bundle]] and [[universal principal ∞-bundle]].)

* Similarly an anafunctor from $P_1(X)$ to $\mathbf{B}G$ is a [[connection on a bundle]] (see there for details).

## Related concepts

* [[smooth groupoid]], [[smooth functor]]

* [[LieGrpd]], [[SmoothGrpd]]

* [[bisection of a Lie groupoid]]

* [[orbifold]]

* [[effective Lie groupoid]], [[proper Lie groupoid]], [[etale groupoid]]

* [[homotopy groups of a Lie groupoid]]

* [[Morita morphism]], [[Hilsum-Skandalis morphism]]

* [[Tannaka duality for Lie groupoids]]

* [[double Lie groupoid]]

* [[higher differential geometry]]

## References

The notion of Lie categories, hence of Lie groupoids, goes back to

* [[Charles Ehresmann]], _Catégories topologiques et categories différentiables_, Colloque de Géométrie différentielle globale, Bruxelles, C.B.R.M., (1959) pp. 137-150 ([[EhresmannCategoriesTopologiques.pdf:file]], [zbMath:0205.28202](https://zbmath.org/?q=an:0205.28202))

Their understanding as [[internal categories]]/[[internal groupoids]] [[internalization|in]] [[SmoothManifolds]] is often attributed to

* [[Charles Ehresmann]], _Catégories structurées_, Annales scientifiques de l'École Normale Supérieure, Série 3, Tome 80 (1963) no. 4, pp. 349-426 ([numdam:ASENS_1963_3_80_4_349_0](http://www.numdam.org/item/ASENS_1963_3_80_4_349_0))

but the simple notion of [[internalization]] and [[internal groupoids]] ([Grothendieck 1960](internalization#Grothendieck60), [61](internalization#Grothendieck61)) is hardly recognizable in this account.

Textbook accounts:

* [[Kirill Mackenzie]], _Lie groupoids and Lie algebroids in differential geometry_, London Mathematical Society Lecture Note Series, 124. Cambridge University Press, Cambridge, 1987. xvi+327 pp ([doi:10.1017/CBO9780511661839](https://doi.org/10.1017/CBO9780511661839), [MR:896907](http://www.ams.org/mathscinet-getitem?mr=896907))

* {#MoerdijkMrcun03} [[Ieke Moerdijk]], [[Janez Mrčun]] _Introduction to Foliations and Lie Groupoids_, Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, 2003  ([doi:10.1017/CBO9780511615450](https://doi.org/10.1017/CBO9780511615450))

  > (in the context of [[foliation theory]]: [[foliation groupoids]])

* [[Kirill Mackenzie]], _General Theory of Lie Groupoids and Lie Algebroids,_ Cambridge University Press, 2005 ([doi:10.1017/CBO9781107325883](https://doi.org/10.1017/CBO9781107325883))

Historical review:

* [[Jean Pradines]], _In [[Ehresmann]]'s footsteps: from Group Geometries to Groupoid Geometries_, Banach Center Publications, vol. 76, Warsawa 2007, 87-157 ([arXiv:0711.1608](http://arxiv.org/abs/0711.1608), [doi:10.4064/bc76-0-5 ](https://www.impan.pl/en/publishing-house/banach-center-publications/all/76/0/86184/in-ehresmann-s-footsteps-from-group-geometries-to-groupoid-geometries))

The relation to [[differentiable stacks]] is discussed/reviewed in section 2 of

* {#Blohmann} [[Christian Blohmann]], _Stacky Lie groups_, Int. Mat. Res. Not. (2008) Vol. 2008: article ID rnn082 ([arXiv:math/0702399](http://arxiv.org/abs/math/0702399))

Lie groupoids as a source for [[groupoid convolution algebras|groupoid convolution]] [[C*-algebras]] are discussed in

* [[Alain Connes]], _[[Noncommutative Geometry]]_

Expository discussion of various kinds of groupoids is also in

* [[John Baez]]  _[TWF](http://math.ucr.edu/home/baez/TWF.html) [256](http://math.ucr.edu/home/baez/week256.html)_.

Groupoids and their various morphisms between them in different categories, including in Diff, is also in

* Ralf Meyer and [[Chenchang Zhu]], _Groupoids in categories with pretopology_,  [arXiv:math/1408.5220](http://arxiv.org/pdf/1408.5220.pdf)

[[!redirects Lie groupoids]]


Lie groupoids > history

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


This entry collects material related to the textbook:

* [[José de Azcárraga]], [[José M. Izquierdo]]:

  \linebreak

  **Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics**

  \linebreak

  Cambridge Monographs of Mathematical Physics

  Cambridge University Press

  (1995)

  [doi:10.1017/CBO9780511599897](https://doi.org/10.1017/CBO9780511599897)

on [[Lie groups]], [[Lie algebras]], [[Lie algebra cohomology]] and applications to [[physics]], such as to the definition of the [[nonrelativistic particle]] regarded as a [[1d WZW model]] and more generally to the definition of [[super p-brane]] [[sigma-models]] as [[higher WZW models]] associated with the "[[brane scan]]" exceptional cocycles of [[super Minkowski spacetimes]].

category: reference


Lie ideal

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## Definition

Given a [[Lie algebra]] with underlying [[vector space]] $\mathfrak{g}$ and [[Lie bracket]] $[-,-]\colon \mathfrak{g} \otimes \mathfrak{g} \to \mathfrak{g}$, then an _[[ideal]]_ is a sub-vector space $V \subset \mathfrak{g}$ such that

$$
  [V,\mathfrak{g}] \subset V
$$

hence such that for all $v \in V \subset \mathfrak{g}$ and $x \in \mathfrak{g}$ one has $[v,x] \in V \subset \mathfrak{g}$.

This is sometimes referred to as a _Lie ideal_.

## Properties

A Lie ideal is always a [[subobject|sub]]-[[Lie algebra]]

## References

See also:

*  [[Groupprops]], _[Ideal of a Lie ring](https://groupprops.subwiki.org/wiki/Ideal_of_a_Lie_ring)_

[[!redirects Lie ideals]]

[[!redirects ideal in a Lie algebra]]
[[!redirects ideals in a Lie algebra]]

[[!redirects ideals in Lie algebras]]


Lie infinity-algebra > history

< [[Lie infinity-algebra]]
[[!redirects Lie infinity-algebra -- history]]