Lie algebra

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## Idea

A _Lie algebra_ is the [[infinitesimal object|infinitesimal]] approximation to a [[Lie group]].

## Definition

### Ordinary definition

A _Lie algebra_ is a [[vector space]] $\mathfrak{g}$ equipped with a bilinear skew-symmetric map $[-,-]  : \mathfrak{g} \wedge \mathfrak{g} \to \mathfrak{g}$ which satisfies the [[Jacobi identity]]:

$$
  \forall x,y,z \in \mathfrak{g} : \left[x,\left[y,z\right]\right] + \left[z,\left[x,y\right]\right] + \left[y,\left[z,x\right]\right] = 0
  \,.
$$

A [[homomorphism]] of Lie algebras is a linear map $\phi : \mathfrak{g} \to \mathfrak{h}$ such that for all $x,y \in \mathfrak{g}$ we have

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

This defines the [[category]] [[LieAlg]] of Lie algebras.

### Internal to a general linear category
 {#InAGeneralLinearCategory}

The notion of _[[Lie algebra]]_ may be formulated [[internalization|internal to]] any [[linear category]]. This general definition subsumes variants of Lie algebras such as [[super Lie algebras]].

Consider a [[commutative unital ring]] $k$, and a (strict for simplicity) [[symmetric monoidal category|symmetric monoidal]] $k$-[[linear category]] $(\mathcal{C},\otimes,1)$ with [[braiding]] $\tau$.

A **[[Lie algebra object]]** in $(\mathcal{C},\otimes,1,\tau)$ is

1. an [[object]]

   $$
     L \in \mathcal{C}
   $$

1. [[morphism]] (the [[Lie bracket]])

   $$
     [-,-] \;\colon\; L \otimes L \to L
   $$

such that the following conditions hold:

1. [[Jacobi identity]]:

   $$
     \left[-,\left[-,-\right]\right]
       +
     \left[-,\left[-,-\right]\right]
     \circ(id_L\otimes\tau_{L,L})
     \circ(\tau\otimes id_L)
     +
     \left[-,\left[-,-\right]\right]
       \circ
     (\tau_{L,L}\otimes id_L)\circ (id_L\otimes\tau_{L,L}) = 0
   $$

1. skew-symmetry:

   $$
     \begin{aligned}
       &
       \phantom{+}
       [-,-]
       \\
       &
       +
       [-,-]\circ \tau_{L,L}
       \\
       & = \phantom{+} 0
     \end{aligned}
   $$

Equivalently, Lie algebra objects  are the [[algebras over an operad]] over a certain quadratic [[operad]], called the [[Lie operad]], which is the [[Koszul dual]] of the [[commutative algebra operad]].

Examples of types of [[Lie algebra objects]]:

If $k$ is the ring $\mathbb{Z}$ of [[integers]] and $\mathcal{C} = $ $k$[[Mod]] = [[Ab]] is the [[category]] of [[abelian groups]] equipped with the [[tensor product of abelian groups]], then a Lie algebra object is called a _[[Lie ring]]_.

If $k$ is a [[field]] and $\mathcal{C} =$ [[Vect]] is the [[category]] of [[vector spaces]] over $k$ equipped with the [[tensor product of vector spaces]] then a Lie algebra object is an ordinary_[[Lie algebra|Lie k-algebra]]_.

If $k$ is a [[field]] and $\mathcal{C}$ = [[sVect]] is the [[category]] of [[super vector spaces]] over $k$, then a Lie algebra object is a  _[[super Lie algebra]]_.

### General abstract perspective
 {#GeneralAbstractPerspective}

Lie algebras are equivalently groups in "[[infinitesimal space|infinitesimal]] geometry".

For instance in [[synthetic differential geometry]] then a Lie algebra of a [[Lie group]] is just the first-order infinitesimal neighbourhood of the unit element (e.g. [Kock 09, section 6](#Kock09)).

More generally in geometric [[homotopy theory]], Lie algebras, being 0-truncated [[L-∞ algebras]] are equivalently "infinitesimal [[∞-group]] [[geometric ∞-stacks]]" (e.g. [here](infinitesimal+cohesive+(infinity,1)-topos#FormalModuliProblems)), also called [[formal moduli problems]] (see there for more).

## Extra stuff, structure, properties

Notions of Lie algebras with extra [[stuff, structure, property]] includes

* extra property

  * [[abelian Lie algebra]]

  * [[simple Lie algebra]]

  * [[semisimple Lie algebra]]

  * [[reductive Lie algebra]]

* extra structure

  * [[super Lie algebra]]

* extra stuff

  * [[∞-Lie algebra]]

    * [[simplicial Lie algebra]]

  * [[∞-Lie algebroid]]

## Properties

### General

* [[PBW theorem]]

### Cohomology

See [[Lie algebra cohomology]].

### Lie theory

See

* [[Lie theory]]

  * [[Lie integration]]

  * [[Lie's three theorems]]

## Examples

* [[translation Lie algebra]], [[line Lie algebra]]

* [[derivation Lie algebra]]

* [[endomorphism Lie algebra]]

* [[general linear Lie algebra]]

* [[orthogonal Lie algebra]]

* [[special orthogonal Lie algebra]]

* [[special unitary Lie algebra]]

* [[Poincare Lie algebra]]

* [[super Poincare Lie algebra]]

## Related concepts

* **Lie algebra**

  * [[abelian Lie algebra|abelian]], [[semisimple Lie algebra|semisimple]], [[reductive Lie algebra|reductive]]

  * [[metric Lie algebra]]

  * [[universal enveloping algebra]], [[Cartan subalgebra]]

  * [[root (in representation theory)]], [[weight (in representation theory)]]

  * [[Lie algebra representation]]

  * [[Lie ideal]]

  * [[Chevalley basis]]

  * [[Kac-Moody algebra]]

  * [[complex Lie algebra]]

  * [[restricted Lie algebra]]

  * [[Lie group]]

  * [[Lie-Poisson structure]]

  * [[Lie algebra extension]]

    * [[semidirect product Lie algebra]]

  * [[Lie algebra contraction]]

* [[Leibniz algebra]]

* [[Lie 2-algebra]], [[Lie 2-group]]

* [[L-∞ algebra]], [[smooth ∞-group]]

* [[Lie algebroid]], [[Lie groupoid]]

* [[L-∞ algebroid]], [[smooth ∞-groupoid]]

* [[Lie coalgebra]]

[[!include infinitesimal and local - table]]

## References

Textbook accounts:

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

* [[Gerhard P. Hochschild]], *Basic Theory of Algebraic Groups and Lie Algebras*, Graduate Texts in Mathematics **75**, Springer (1981) [[doi:10.1007/978-1-4613-8114-3_16](https://doi.org/10.1007/978-1-4613-8114-3_16)]

* [[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]])

* {#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

* [[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]], 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)]

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

* {#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))

* [[Peter Woit]], Ch. 5 of *Quantum Theory, Groups and Representations: An Introduction*, Springer 2017 [[doi:10.1007/978-3-319-64612-1](https://doi.org/10.1007/978-3-319-64612-1), ISBN:978-3-319-64610-7]

Discussion with a view towards [[Chern-Weil theory]]:

* [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], chapter IV in vol III of: _[[Connections, Curvature, and Cohomology]]_ Academic Press (1973)

Discussion in [[synthetic differential geometry]] is in

* {#Kock09} [[Anders Kock]], section 6 of _Synthetic Geometry of Manifolds_, 2009 ([pdf](http://home.imf.au.dk/kock/SGM-final.pdf))

[[!redirects Lie algebras]]

[[!redirects Lie bracket]]
[[!redirects Lie brackets]]


Lie algebra cohomology

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#### Lie theory
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[[!include infinity-Lie theory - contents]]
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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

_Lie algebra cohomology_ is the intrinsic notion of [[cohomology]] of [[Lie algebra]]s.

There is a precise sense in which Lie algebras $\mathfrak{g}$ are [[infinitesimal object|infinitesimal]] [[Lie group]]s. Lie algebra cohomology is the restriction of the definition of [[Lie group cohomology]] to Lie algebras.

In [[∞-Lie theory]] one studies the relation between the two via [[Lie integration]].

Lie algebra cohomology generalizes to [[nonabelian Lie algebra cohomology]] and to [[∞-Lie algebra cohomology]].

## Definition

There are several different but equivalent definitions of the [[cohomology]] of a [[Lie algebra]].

### As Ext-group or derived functor

The abelian [[cohomology]] of a $k$-[[Lie algebra]] $\mathfrak{g}$ with coefficients in the left $\mathfrak{g}$-module $M$ is defined as $H^*_{Lie}(\mathfrak{g},M) = Ext_{U\mathfrak{g}}^*(k,M)$ where $k$ is the ground field understood as a trivial module over the universal enveloping algebra $U\mathfrak{g}$. In particular it is a derived functor.

### Via resolutions

Before this approach was advanced in Cartan-Eilenberg's _Homological algebra_, Lie algebra cohomology and homology were defined by Chevalley-Eilenberg with a help of concrete Koszul-type resolution which is in this case a cochain complex

$$Hom_{\mathfrak{g}}(U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g},M)\cong Hom_k(\Lambda^* \mathfrak{g},M),$$

where the first argument $U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g}$ is naturally equipped with a differential to start with (see below).

WHERE  BELOW?

 The first argument in the Hom, i.e. $U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g}$ is sometimes called the Chevalley-Eilenberg chain complex (cf. Weibel); the [[Chevalley-Eilenberg cochain complex]] is the whole thing, i.e.

$$CE(\mathfrak{g},M) := Hom_{\mathfrak{g}}(U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g},M)\cong Hom_k(\Lambda^* \mathfrak{g},M).$$

If $M$ is a trivial module $k$ then $CE(\mathfrak{g}) := Hom_k(\Lambda^* \mathfrak{g},k)$ and if $\mathfrak{g}$ is finite-dimensional this equals $\Lambda^* \mathfrak{g}^*$ with an appropriate differential and the exterior multiplication gives it a dg-algebra structure.

### Via $\infty$-Lie algebras

As discussed at [[Chevalley-Eilenberg algebra]], we may identify [[Lie algebra]]s $\mathfrak{g}$ as the duals $CE(\mathfrak{g})$ of [[dg-algebra]]s whose underlying graded algebra is the [[Grassmann algebra]] on the vector space $\mathfrak{g}^*$.

Similarly, a dg-algebra $CE(\mathfrak{h})$ whose underlying algebra is free on a [[graded vector space]] $\mathfrak{h}$ we may understand as exibiting an [[∞-Lie algebra]]-structure on $\mathfrak{h}$.

Then a morphism $\mathfrak{g} \to \mathfrak{h}$ of these $\infty$-Lie algebras is by definition just a morphism $CE(\mathfrak{g}) \leftarrow CE(\mathfrak{h})$ of dg-algebras. Such a morphis may be thought of as a cocycle in [[nonabelian Lie algebra cohomology]] $H(\mathfrak{g}, \mathfrak{h})$.

Specifically, write $b^{n-1} \mathbb{R}$ for the [[line Lie n-algebra]], the  $\infty$-Lie algebra given by the fact that $CE(b^{n-1}\mathbb{R})$ has a single generator in degree $n$ and vanishing differential. Then a morphism

$$
  \mu : \mathfrak{g} \to b^{n-1} \mathbb{R}
$$

is a cocycle in the abelian Lie algebra cohomology $H^n(\mathfrak{g}, \mathbb{R})$. Notice that dually, by definition, this is a morphism of dg-algebras

$$
  CE(\mathfrak{g}) \leftarrow CE(b^{n-1} \mathbb{R}) : \mu
  \,.
$$

Since on the right we only have a single closed degree-$n$ generator, such a morphism is precily a closed degree $n$-element

$$
  \mu \in CE(\mathfrak{g})
  \,.
$$

This way we recover the above definition of Lie algebra cohomology (with coefficient in the trivial module) in terms of the cochain complex cohomology of the CE-algebra.

## Properties

### Whitehead's lemma

The following lemma asserts that for semisimple Lie algebras $\mathfrak{g}$ only the cohomology $\mathfrak{g} \to b^{n-1} \mathbb{R}$ with coefficients in the trivial module is nontrivial.

+-- {: .num_prop}
###### Proposition
**(Whitehead's lemma)**

For $\mathfrak{g}$ a finite dimensional [[semisimple Lie algebra]] over a [[field]] of [[characteristic]] 0, and for $V$ a non-trivial finite-dimensional [[irreducible representation]], we have

$$
  H^p(\mathfrak{g}, V) = 0 \;\;\; for\;p \gt 0
  \,.
$$

=--

### Van Est isomorphism

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.

### Hochschild-Serre spectral sequence
  {#HochschildSerreSpectralSequence}

+-- {: .num_defn #RelativeLieAlgebraCohomologyWithCoefficients}
###### Definition
**(relative Lie algebra cohomology with coefficents)**

Let

1. $(\mathfrak{g}, [-,-])$ be a [[Lie algebra]] of [[finite number|finite]] [[dimension]];

1. $(V, \rho)$ a $\mathfrak{g}$-[[Lie algebra module]] of [[finite number|finite]] [[dimension]];

1. $\mathfrak{h} \hookrightarrow \mathfrak{g}$ a sub-Lie algebra.

Consider the $\mathbb{N}$-[[graded vector space]]

$$
  C^\bullet(\mathfrak{g}, \mathfrak{h}, V)
  \;\coloneqq\;
    \left(
      (
        \wedge^\bullet
        (\mathfrak{g}/\mathfrak{h})^\ast
      )
      \otimes V
    \right)^{\mathfrak{h}}
$$

consisting of the $\mathfrak{h}$-[[invariant|invariant elements]] in the [[tensor product]] of $V$ with the [[exterior algebra]] of the [[coset]] $\mathfrak{g}/\mathfrak{h}$.

On this graded vector space, the [[dual linear maps]] of the [[Lie bracket]] $[-,-]$, extended as a graded [[derivation]] to the exterior algebra, and the [[Lie algebra action]] $\rho$ define a [[differential]]

$$
  d_{CE} \coloneqq \rho^\ast  + [-,-]^\ast
  \,.
$$

The resulting [[cochain complex]] is the  _[[Chevalley-Eilenberg complex]] of $\mathfrak{g}$ relative $\mathfrak{h}$ with [[coefficients]] in $V$_.

Its [[cochain cohomology]]

$$
  H^\bullet(\mathfrak{g}, \mathfrak{h}; V)
  \;\coloneqq\;
  \left(
    \left(
      (
        \wedge^\bullet
        (\mathfrak{g}/\mathfrak{h})^\ast
      )
      \otimes V
    \right)^{\mathfrak{h}}
    ,
    d_{CE}
  \right)
$$

is the _Lie algebra cohomology of $\mathfrak{g}$ relative $\mathfrak{h}$ with [[coefficients]] in $V$_.

=--

(e.g. [Solleveld 02, def. 2.13 and def. 2.17](#Solleveld02))

If in def. \ref{RelativeLieAlgebraCohomologyWithCoefficients} $\mathfrak{h} = 0$ then the definition reduces to that of ordinary Lie algebra cohomology with coefficients:

$$
  H^\bullet(\mathfrak{g}, 0; V)
  =
  H^\bullet(\mathfrak{g}; V)
  \,.
$$

+-- {: .num_defn #LieAlgebraReductiveInAmbientLieAlgebra}
###### Definition
**([[reductive Lie algebra|Lie algebra reductive]] in ambient Lie algebra)**

A sub-[[Lie algebra]]

$$
  \mathfrak{h} \hookrightarrow \mathfrak{g}
$$

is called _reductive_ if the [[adjoint representation|adjoint]] [[Lie algebra representation]] of $\mathfrak{h}$ on $\mathfrak{g}$ is [[reducible representation|reducible]].

=--

([Koszul 50](#Koszul50), recalled in e.g. [Solleveld 02, def. 2.27](#Solleveld02))

+-- {: .num_prop #InvariantsInLieAlgebraCohomologyComputedByRelativeLieAlgebraCohomology}
###### Proposition
**(invariants in Lie algebra cohomology computed by relative Lie algebra cohomology)**

Let

1. $(\mathfrak{g}, [-,-])$ be a [[Lie algebra]] of [[finite number|finite]] [[dimension]];

1. $(V, \rho)$ a $\mathfrak{g}$-[[Lie algebra module]] of [[finite number|finite]] [[dimension]], which is [[reducible representation|reducible]];

1. $\mathfrak{h} \hookrightarrow \mathfrak{g}$ a sub-Lie algebra which is [[reductive Lie algebra|reductive]] in $\mathfrak{g}$ (Def. \ref{LieAlgebraReductiveInAmbientLieAlgebra}) in that its [[adjoint representation]] on $\mathfrak{g}$ is [[reducible representation|reducible]].

1. such that

   $$
     \mathfrak{g} = \mathfrak{h} \ltimes \mathfrak{a}
   $$

   is a [[semidirect product Lie algebra]] (hence $\mathfrak{a}$ a [[Lie ideal]]).

Then the [[invariants]] in the Lie algebra cohomology of $\mathfrak{a}$ (either with respect to $\mathfrak{h}$ or all of $\mathfrak{g}$) coincide with the relative Lie algebra cohomology (Def. \ref{RelativeLieAlgebraCohomologyWithCoefficients}, using the invariant subcomplex!):

$$
   H^\bullet(\mathfrak{a}; V)^{\mathfrak{h}}
   \;\simeq\;
   H^\bullet(\mathfrak{g}, \mathfrak{h}; V)
   \,.
$$

=--

([Solleveld 02, theorem 2.28](#Solleveld02))

Proof via the [[Hochschild-Serre spectral sequence]].

## Examples

Every [[invariant polynomial]] $\langle - \rangle \in W(\mathfrak{g})$ on a Lie algebra has a _transgression_ to a cocycle on $\mathfrak{g}$. See [[∞-Lie algebra cohomology]] for more.

For instance for $\mathfrak{g}$ a [[semisimple Lie algebra]], there  is the [[Killing form]] $\langle - ,- \rangle$. The corresponding 3-cocycle is

$$
  \mu \;\coloneqq\;
  \big\langle -, [-,-] \big\rangle
  \;\in\;
  CE(\mathfrak{g})
  \,,
$$

that is: the function that sends a [[tripe]] of Lie algebra elements $x, y, z$ to the number  $\mu(x,y,z) = \big\langle x, [y,z]\big\rangle$.

On the [[super Poincare Lie algebra]] in dimension (10,1) there is a 4-cocycle

$$
  \mu_4 = \bar \psi \wedge \Gamma^{a b} \Psi\wedge e_a \wedge e_b
  \in
  CE(\mathfrak{siso}(10,1))
$$

## Extensions

Every Lie algebra degree $n$ cocycle $\mu$ (with values in the trivial model) gives rise to an extension

$$
  b^{n-2} \mathbb{R} \to \mathfrak{g}_{\mu}
  \to \mathfrak{g}
  \,.
$$

In the language of [[∞-Lie algebra]]s this was observed in ([BaezCrans Theorem 55](#BaezCrans)).

In the dual [[dg-algebra]] language the extension is lust the relative [[Sullivan algebra]]

$$
  CE(\mathfrak{g}_\mu) \leftarrow CE(\mathfrak{g})
$$

obtained by gluing on a rational $n$-sphere. By this kind of translation between familiar statements in [[rational homotopy theory]] dually into the language of [[∞-Lie algebra]]s many useful statements in [[∞-Lie theory]] are obtained.

**Examples**

* The [[string Lie 2-algebra]] is the extension of a [[semisimple Lie algebra]] induced by the canonical 3-cocycle coming from the [[Killing form]].

* The [[supergravity Lie 3-algebra]] is the extension of the [[super Poincare Lie algebra]] by a 4-cocycle.

## Related concepts

* [[Hochschild-Serre spectral sequence]]

* [[signs in supergeometry]]

* [[infinity-Lie algebra cohomology]]

## References

### Ordinary Lie algebras

Accounts of the standard theory of Lie algebra cohomology include

* {#Koszul50} [[Jean-Louis Koszul]], _Homologie et cohomologie des algèbres de Lie_, Bull. Soc. Math. France 78 (1950), 65-127

* [[Gerhard Hochschild]], [[Jean-Pierre Serre]], _Cohomology of Lie algebras_, Annals of Mathematics, Second Series, Vol. 57, No. 3 (May, 1953), pp. 591-603 ([JSTOR](http://www.jstor.org/stable/1969740))

* [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], in chapter V in vol III of _[[Connections, Curvature, and Cohomology]]_ Academic Press (1973)

* [[Charles Weibel]], chapter 7 of _An introduction to homological algebra_, Cambridge Studies in Adv. Math. __38__, CUP 1994

* [[José de Azcárraga]], [[José M. Izquierdo]], Section 6 of: *[[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)]

* [[Michael Barr]], *Cartan-Eilenberg cohomology and triples*, J. Pure Applied Algebra **112** 3 (1996) 219–238 [doi:10.1016/0022-4049(95)00138-7, [pdf](https://www.math.mcgill.ca/barr/papers/coho.pdf), [[Barr-CECohomology.pdf:file]]]

* [[José de Azcárraga]], [[José M. Izquierdo]], J. C. Perez Bueno, *An introduction to some novel applications of Lie algebra cohomology and physics*, RACSAM **95** 2 (2001) 225-248 [[arXiv:physics/9803046](http://arxiv.org/abs/physics/9803046), [eudml:40921](https://eudml.org/doc/40921)]

* {#Solleveld02} [[Maarten Solleveld]], _Lie algebra cohomology and Macdonald’s conjectures_, 2002 ([pdf](https://www.math.ru.nl/~solleveld/scrip.pdf))

See also

* [[eom]] _[Lie algebra cohomology](http://eom.springer.de/C/c023140.htm)_

* [wikipedia](http://en.wikipedia.org/wiki/Lie_algebra_cohomology)

* scholarpedia: [An introduction to Lie algebra cohomology](http://www.scholarpedia.org/article/An_introduction_to_Lie_algebra_cohomology)

### Super Lie algebras
 {#ReferencesSuperLieAlg}

The cohomology of [[super Lie algebra]]s is analyzed via [[normed division algebra]]s in

* [[John Baez]], [[John Huerta]], _Division algebras and supersymmetry I_ ([arXiv:0909.0551](http://arxiv.org/abs/0909.0551))

* [[John Baez]], [[John Huerta]], _Division algebras and supersymmetry II_ ([arXiv:1003.34360](http://arxiv.org/abs/1003.3436))

See also [[division algebra and supersymmetry]].

This subsumes some of the results in

* [[José de Azcárraga]], [[Paul Townsend]], _Superspace geometry and classification of supersymmetric extended objects_, Phys. Rev. Lett. 62, 2579--2582 (1989) ([doi:10.1103/PhysRevLett.62.2579](https://doi.org/10.1103/PhysRevLett.62.2579))

The cohomology of the [[super Poincare Lie algebra]] in low dimensions $\leq 5$ is analyzed in

* [[Friedemann Brandt]],

  _Supersymmetry algebra cohomology I: Definition and general structure_ 	J. Math. Phys.51:122302, 2010, [arXiv](http://arxiv.org/abs/0911.2118)

  _Supersymmetry algebra cohomology II: Primitive elements in 2 and 3 dimensions_ 	J. Math. Phys. 51 (2010) 112303 ([arXiv](http://arxiv.org/abs/1004.2978))

  _Supersymmetry algebra cohomology III: Primitive elements in four and five dimensions_ ([arXiv](http://arxiv.org/abs/1005.2102))

and in higher dimensions more generally in

* Michael Movshev, [[Albert Schwarz]], Renjun Xu, _Homology of Lie algebra of supersymmetries_ ([arXiv](http://arxiv.org/abs/1011.4731)) .

On [[Lie algebra cohomology]] of [[super Lie algebras]] (see also the [[Green-Schwarz action functional|brane scan]]) in relation to [[integration over supermanifolds|integrable forms]] of coset [[supermanifolds]]:

* [[Roberto Catenacci]], [[C. A. Cremonini]], [[Pietro Antonio Grassi]], [[Simone Noja]], _Cohomology of Lie Superalgebras: Forms, Integral Forms and Coset Superspaces_ ([arXiv:2012.05246](https://arxiv.org/abs/2012.05246))

* [[C. A. Cremonini]], [[Pietro Antonio Grassi]], *Cohomology of Lie Superalgebras: Forms, Pseudoforms, and Integral Forms* ([arXiv:2106.11786](https://arxiv.org/abs/2106.11786))

### Extensions

The [[∞-Lie algebra]] [[∞-Lie algebra cohomology|extensions]] $b^{n-2} \to \mathfrak{g}_\mu \to \mathfrak{g}$ induced by a degree $n$-cocycle are considered

* {#BaezCrans} [[John Baez]], [[Alissa Crans]], around theorem 55 in _Higher-Dimensional Algebra VI: Lie 2-Algebras_, Theory and Applications of Categories 12 (2004), 492-528.  [arXiv](http://arxiv.org/abs/math.QA/0307263)

[[!redirects Lie algebra cocycle]]
[[!redirects Lie algebra cocycles]]


Lie algebra contraction

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[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a Lie algebra $\mathfrak{g}$ over the [[real numbers]] or [[complex numbers]], and gives a choice $\{t_0, t_a\}_{a \in I}$ of [[linear basis]] of $\mathfrak{g}$, the corresponding _Inönü-Wigner contraction_ is the Lie algebra obtained by "sending $t_0$ to zero", i.e. the Lie algebra obtained from the previous one by passing to basis elements $\{\epsilon t_0, t_a\}_{a \in I}$ with $\epsilon$ in the ground field, in the [[limit of a sequence|limit]] that $\epsilon \to 0$.

## Related concepts

* [[Penrose limit]]

## References

The original article is

* [[Erdal İnönü]], [[Eugene Wigner]] (1953). _On the Contraction of Groups and Their Representations_. Proc. Nat. Acad. Sci. 39 (6): 510–24.

See also

* Wikipedia, _[Group contraction](http://en.wikipedia.org/wiki/Group_contraction)_

*  E. J. Saletan,  _Contraction of Lie Groups_. Journal of Mathematical Physics 2: 1–1 (1961)

[[!redirects Lie algebra contractions]]

[[!redirects group contraction]]
[[!redirects group contractions]]

[[!redirects Lie group contraction]]
[[!redirects Lie group contractions]]

[[!redirects İnönü-Wigner contraction]]
[[!redirects İnönü-Wigner contractions]]

[[!redirects Inönü-Wigner contraction]]
[[!redirects Inönü-Wigner contractions]]

[[!redirects Inonu-Wigner contraction]]
[[!redirects Inonu-Wigner contractions]]

[[!redirects Inönü-Wigner group contraction]]
[[!redirects Inönü-Wigner group contractions]]


Lie algebra extension

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#### Cohomology
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#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

An _extension_ of a [[Lie algebra]] $\mathfrak{g}$ is another Lie algebra $\hat {\mathfrak{g}}$ that is equipped with a surjective Lie algebra homomorphism to $\mathfrak{g}$

$$
  \array{
     \hat{\mathfrak{g}}
     \\
     \downarrow
     \\
     \mathfrak{g}
  }
  \,.
$$

For non-trivial extensions, this homomorphism has a [[kernel]] $\mathfrak{a} \hookrightarrow \hat \mathfrak{g}$ , consisting of those elements of $\hat{\mathfrak{g}}$ that map to the zero element in $\mathfrak{g}$. That kernel is a sub-Lie algebra of $\hat{\mathfrak{g}}$ and hence one says that $\hat\mathfrak{g}$ is an extension of $\mathfrak{g}$ by $\mathfrak{a}$.

$$
  \array{
     \mathfrak{a} &\hookrightarrow& \hat{\mathfrak{g}}
     \\
     &&\downarrow
     \\
     && \mathfrak{g}
  }
  \,.
$$

This means equivalently that there is a [[short exact sequence]] of Lie algebras of the form

$$
  0 \to \mathfrak{a} \longrightarrow \hat \mathfrak{g} \longrightarrow \mathfrak{g} \to 0
  \,.
$$

When $\mathfrak{a}$ happens to be abelian, hence when its Lie bracket is trivial, then one speaks of an abelian extension, and when furthermore the Lie bracket of $\hat\mathfrak{g}$ vanishes as soon as already one of its arguments is in $\mathfrak{a}$, then one has a [[central extension]] ($\mathfrak{a}$ is in the [[center]] of $\hat \mathfrak{g}$).

Central extensions by the [[ground field]] (say $\mathbb{R}$) are equivalently induced by a 2-cocycle $\mu_2$ in the [[Lie algebra cohomology]] of $\mathfrak{g}$ with [[coefficients]] in the [[ground field]], say $\mathbb{R}$, i.e. by linear maps

$$
  \mu_2 \colon \mathfrak{g} \wedge \mathfrak{g} \longrightarrow \mathbb{R}
$$

satisfying some conditions. The corresponding extension of $\mathfrak{g}$ is then, at the level of underlying vector space, the [[direct sum]] $\hat \mathfrak{g} = \mathfrak{g} \oplus \mathbb{R}$, equipped with the Lie bracket given by the formula

$$
  [(x_1,t_1), (x_2,t_2)] = ([x_1,x_2], \mu_2(x_1,x_2))
$$

for all $x_1,x_2 \in \mathfrak{g}$ and $t_1,t_2 \in \mathbb{R}$. The condition on $\mu_2$ to be a 2-cocycle is precisely the condition that this formula satisfies the [[Jacobi identity]].

If one regards all Lie algebras here as being special cases of [[Lie 2-algebras]], then the 2-cocycle $\mu_2$ may itself be thought of as a homomorphism, namely from $\mathfrak{g}$ to the [[line Lie 2-algebra]] $b\mathbb{R}$. With this, then $\hat \mathfrak{g}$ given by the above formula is simply the [[homotopy fiber]] of $\mu_2$, and the whole story comes down to saying that there is a [[homotopy fiber sequence]] of [[L-∞ algebras]] of the form

$$
  \array{
     \mathbb{R} &\hookrightarrow& \hat{\mathfrak{g}}
     \\
     &&\downarrow
     \\
     && \mathfrak{g}
     &\stackrel{\mu_2}{\longrightarrow}& b \mathbb{R}
  }
  \,.
$$

This perspective on Lie algebra extensions makes it evident how the concept generalizes to a concept of [L-∞ algebra extensions](infinity-Lie+algebra+cohomology#Extensions).

Of course extensions need not be central or even abelian. An important class of non-abelian extensions are [[semidirect product Lie algebras]]. These are given by an Lie [[action]] of $\mathfrak{g}$ on $\mathfrak{a}$, hence a [[homomorphism]] $\rho \colon \mathfrak{g}\longrightarrow \mathfrak{der}(\mathfrak{a})$ to the [[derivations]] on $\mathfrak{a}$ and with this the bracket on $\mathfrak{g} \oplus \mathfrak{a}$ is given by the formula

$$
  [(x_1,t_1), (x_2,t_2)]
  =
  ( [x_1,x_2], \;([t_1,t_2] + \rho(x_1)(t_2) - \rho(x_2)(t_1)) )
  \,.
$$

## Definition

A [[short exact sequence]] of [[Lie algebras]] is a [[diagram]]

$$ 0\to \mathfrak{k}
\overset{i}\to \mathfrak{g}\overset{p}\to\mathfrak{b}\to 0$$

where $\mathfrak{k},\mathfrak{g},\mathfrak{b}$ are Lie algebras, $i,p$ are homomorphisms of Lie algebras and the underlying diagram of vector spaces is exact, i.e. $Ker(p)=Im(i)$, $Ker(i)=0$ and $Im(p)=\mathfrak{b}$.

We also say that this diagram (and sometimes, loosely speaking, $\mathfrak{g}$ itself) is a **Lie algebra extension** of $\mathfrak{b}$ by the "kernel" $\mathfrak{k}$.

Lie algebra extensions may be obtained from [[Lie group]] [[group extensions]] via the tangent Lie algebra functor.

## Properties

### Classification by nonabelian Lie algebra cohomology

We discuss how in general Lie algebra extensions are classified by cocycles in [[nonabelian Lie algebra cohomology]].

Each element $g \in \mathfrak{g}$ defines
a derivative $\phi(g)$ on $\mathfrak{k}$ by $\phi(g)(k) = [g,k]$.
The rule $g \mapsto \phi(g)$ defines a homomorphism of Lie algebras
$\phi : \mathfrak{g} \rightarrow Der(\mathfrak{k})$.
Indeed,

$$
\begin{aligned}
\phi([g_1,g_2])(k) &= [[g_1,g_2],k] =
[[g_1,k],g_2] + [g_1,[g_2,k]] = -\phi(g_2)([g_1,k]) + \phi(g_1)([g_2,k])
=\\
&= [-\phi(g_2)\circ\phi(g_1) + \phi(g_1)\circ\phi(g_2)](k) = [\phi(g_1),\phi(g_2)](k),
\end{aligned}
$$

for all $g_1,g_2 \in \mathfrak{g}$, for all $k \in \mathfrak{k}$.
The restriction $\phi|_{\mathfrak{k}}$ takes
(by definition) values in the Lie subalgebra $Int(\mathfrak{k})$ of inner derivatives of $\mathfrak{k}$.
If $g_1$ and $g_2$ are in the same coset, that is
$g_1 + \mathfrak{k} = g_2 + \mathfrak{k}$,
then there is $k \in \mathfrak{k}$ with $g_1 + k = g_2$ and
such that for all $k' \in \mathfrak{k}$ we have
$\phi(g_1) + \phi(k') = \phi(g_1 + k') = \phi(g_2 + k + k') = \phi(g_2)+\phi(k + k')$ and therefore

$$\array{\phi(g_1) + Int(\mathfrak{k}) &=& \phi(g_1) + \phi(\mathfrak{k})\\
&=& \phi(g_1 + \mathfrak{k}) \\
&=& \phi(g_2 + \mathfrak{k}) \\
&=& \phi(g_2) + \phi(\mathfrak{k})\\
&=& \phi(g_2) + Int(\mathfrak{k}).}$$

Thus we obtain a well-defined map $\phi_* : \mathfrak{g}/\mathfrak{k} \to Der(\mathfrak{k})/Int(\mathfrak{k})$.

Choose a $k$-linear section of the projection
$\mathfrak{g} \rightarrow \mathfrak{g}/\mathfrak{k}\cong \mathfrak{b}$
and denote by $\psi$ the composition $\phi \circ \sigma$
where  $\sigma : \mathfrak{g}/\mathfrak{k} = \mathfrak{b} \rightarrow \mathfrak{g}$.
One considers the problem of reconstructing the Lie algebra $\mathfrak{g}$ from
the knowledge of $\psi : \mathfrak{g}/\mathfrak{k} \rightarrow Der(\mathfrak{k})$ and $\mathfrak{k}$.
In order to derive the necessary relations we
will identify $\mathfrak{g}$ with $\mathfrak{b} \times \mathfrak{k}$ (as a set).

Indeed, write each element $g \in \mathfrak{g}$ as
$\sigma(b) + k, b \in \mathfrak{g}/\mathfrak{k}$, $k \in \mathfrak{k}$ by setting $b := [g], k := -\sigma([g]) + g$. The elements $b \in \mathfrak{b}$ and
$k \in \mathfrak{k}$ in this decomposition are unique.
Thus we obtain a bijection $\mathfrak{g} \rightarrow \mathfrak{b} \times \mathfrak{k}$, $g \mapsto ([g], -\sigma([g]) + g )$.
The commutation rule has to be figured out.
If $(b_1,k_1) = g_1$, and $(b_2,k_2) = g_2$, then
\begin{equation}
[g_1,g_2] = [\sigma(b_1) + k_1,\sigma(b_2) + k_2] =
[\sigma(b_1),\sigma(b_2)] + [\sigma(b_1),k_2] - [\sigma(b_2),k_1] +[k_1,k_2].
\label{putkhilie}
\end{equation}

 Now $[\sigma(b_1),\sigma(b_2)] \in [[b_1,b_2]]$ so it can be represented uniquely
in the form $\sigma([b_1,b_2]) + k$ where $k \in \mathfrak{k}$ can be obtained by evaluating the antisymmetric $k$-bilinear form
$\chi : \mathfrak{b} \wedge \mathfrak{b} \rightarrow \mathfrak{k}$ defined by
$\chi(b_1 \wedge b_2) = - \sigma([b_1,b_2]) + [\sigma(b_1),\sigma(b_2)]$
 on $(b_1,b_2)$.
Then formula \eqref{putkhilie} becomes
$$\array{
[g_1,g_2] & = & \sigma([b_1,b_2]) + \chi(b_1\wedge b_2)
+ \phi(\sigma(b_1))(k_2) + \phi(-\sigma(b_2))(k_1) + [k_1,k_2] \\
       & = & \sigma([b_1,b_2]) + \chi(b_1\wedge b_2) + \psi(b_1)(k_2)
        -\psi(b_2)(k_1) + [k_1,k_2].
}$$

so that

\begin{equation}
[(b_1,k_1),(b_2,k_2)] = \big([b_1,b_2],\chi(b_1\wedge b_2) +
        \psi(b_1)(k_2) - \psi(b_2)(k_1) +  [k_1,k_2]\big).
\label{mrulelie}
\end{equation}

Thus all the information about the commutators is encoded
in functions $\chi : \mathfrak{b} \wedge \mathfrak{b} \rightarrow \mathfrak{k}$
and $\psi : \mathfrak{b} \to Der(\mathfrak{k})$,
without knowledge of $\sigma$.

However, not every pair $(\chi,\psi)$ will
give some commutation rule on $\mathfrak{b} \times \mathfrak{k}$ satisfying the Jacobi identity and some different pairs may lead to the isomorphic extensions.

In order to satisfy the Jacobi identity, this pair needs to form a nonabelian 2-cocycle in the sense of [[nonabelian Lie algebra cohomology]].

## Examples

* The [[Heisenberg Lie algebra]] is an extension of $\mathbb{R}^{2}$, regarded as an abelian Lie algebra, by $\mathbb{R}$ with the corresponding 2-cocycle $\mu_2$ being the canonical commutation relation $\mu_2(q,q)= 0$, $\mu_2(p,p)= 0$, $\mu_2(q,p) = 1$.

* More generally, the [[Kostant Souriau extension]] exhibits a [[Poisson bracket]] on a [[symplectic manifold]] as an extension of the Lie algebra of [[Hamiltonian vector fields]].

For more discussion putting these two examples in perspective see also at _[quantization -- Motivation from classical mechanics and Lie theory](quantization#MotivationFromClassicalMechanicsAndLieTheory)_.

## Related concepts

* [[semidirect sum Lie algebra]]

* [[central extension]], [[central charge]]

* [[Lie algebra contraction]]

## References

Discussion in the generality of [[super Lie algebras]] includes

* [[Dmitri Alekseevsky]], [[Peter Michor]], Wolfgang Ruppert, _Extensions of super Lie algebras_, J. Lie Theory 15 (2005) No. 1, 125--134 ([arXiv:math/0101190](http://arxiv.org/abs/math/0101190))

See also

* Wikipedia, _[Lie algebra extension](https://en.wikipedia.org/wiki/Lie_algebra_extension)_

[[!redirects Lie algebra extensions]]
[[!redirects extension of Lie algebras]]
[[!redirects extensions of Lie algebras]]

[[!redirects prolongation]]
[[!redirects prolongations]]


Lie algebra homology

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#### Cohomology
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[[!include cohomology - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

Dual to [[Lie algebra cohomology]]. See there for more.

## Definition

The abelian [[homology]] of a $k$-[[Lie algebra]] $\mathfrak{g}$ with coefficients in the left $\mathfrak{g}$-module $M$ is defined as $H_*^{Lie}(\mathfrak{g},M) = Tor^{U\mathfrak{g}}_*(k,M)$ where $k$ is the ground field understood as a trivial module over the universal enveloping algebra $U\mathfrak{g}$. In particular it is a [[derived functor]]. It can be computed using [[Chevalley-Eilenberg chain complex]] $V(\mathfrak{g})$ as the homology of the chain complex

$$ M \otimes_{U\mathfrak{g}} V(\mathfrak{g}) = M\otimes_{U\mathfrak{g}} U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g} = M\otimes_k \Lambda^* \mathfrak{g}. $$

## Properties

### Loday-Quillen-Tsygan theorem

The _[[Loday-Quillen-Tsygan theorem]]_ ([Loday-Quillen 84](#LodayQuillen84), [Tsygan 83](#Tsygan83)) states that for any [[associative algebra]], $A$ in [[characteristic zero]], the [[Lie algebra homology]] $H_\bullet(\mathfrak{gl}(A))$  of the infinite [[general linear Lie algebra]] $\mathfrak{gl}(A)$ with [[coefficients]] in $A$ is, up to a degree shift, the [[exterior algebra]] $\wedge(HC_{\bullet - 1}(A))$ on the [[cyclic homology]] $HC_{\bullet - 1}(A)$ of $A$:

$$
  H_\bullet(\mathfrak{gl}(A))
  \;\simeq\;
  \wedge( HC_{\bullet - 1}(A) )
$$

(see e.g [Loday 07, theorem 1.1](#Loday07)).

Lecture notes include

* {#Loday07} [[Jean-Louis Loday]], _Cyclic Homology Theory, Part II_, notes taken by Paweł Witkowski (2007) ([pdf](https://www.impan.pl/swiat-matematyki/notatki-z-wyklado~/loday_cht_2.pdf))

## References

* C. Chevalley, S. Eilenberg, _Cohomology theory of Lie groups and Lie algebras_, Trans. Amer. Math. Soc. 63, (1948). 85–124.

The [[Loday-Quillen-Tsygan theorem]] is originally due, independently, to

* {#LodayQuillen84} [[Jean-Louis Loday]], [[Daniel Quillen]], _Cyclic homology and the Lie algebra homology of matrices_ Comment. Math. Helv., 59(4):569–591, 1984.

and

* {#Tsygan83} [[Boris Tsygan]], _Homology of matrix algebras over rings and the Hochschild homology_, Uspeki Math. Nauk., 38:217–218, 1983.


Lie algebra object

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#### Categorical algebra
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[[!include categorical algebra -- contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _[[Lie algebra]]_ may be formulated [[internalization|internal to]] general [[linear category|linear]] [[monoidal categories]] ([[tensor categories]]). This general definition of _Lie algebra objects internal to tensor categories_ subsumes variants of Lie algebras such as [[super Lie algebras]].

## Definition
 {#Definition}

### As internal Lie algebras
 {#AsInternalLieAlgebras}

Consider a [[commutative unital ring]] $k$, and a (strict for simplicity) [[symmetric monoidal category|symmetric monoidal]] $k$-[[linear category]] $(\mathcal{C},\otimes,1)$ with [[braiding]] $\tau$.

A **[[Lie algebra object]]** in $(\mathcal{C},\otimes,1,\tau)$ is

1. an [[object]]

   $$
     L \in \mathcal{C}
   $$

1. [[morphism]] (the [[Lie bracket]])

   $$
     [-,-] \;\colon\; L \otimes L \to L
   $$

such that the following conditions hold:

1. [[Jacobi identity]]:

   $$
     \left[-,\left[-,-\right]\right]
       +
     \left[-,\left[-,-\right]\right]
     \circ(id_L\otimes\tau_{L,L})
     \circ(\tau\otimes id_L)
     +
     \left[-,\left[-,-\right]\right]
       \circ
     (\tau_{L,L}\otimes id_L)\circ (id_L\otimes\tau_{L,L}) = 0
   $$

1. skew-symmetry:

   $$
     \begin{aligned}
       &
       \phantom{+}
       [-,-]
       \\
       &
       +
       [-,-]\circ \tau_{L,L}
       \\
       & = \phantom{+} 0
     \end{aligned}
   $$

Equivalently, in [[string diagram]]-notation in the ambient [[tensor category]], the [[Lie action property]] looks as follows:

$$
  \begin{aligned}
    \Leftrightarrow
    &
    \;\;\;\;\;
    \rho(f(x,y),z) \;=\; \rho(y,\rho(x,z)) - \rho(x,\rho(y,z))
    \\
    \underset{
      {f = [-,-]}
      \atop
      {Lie\;bracket}
    }{
      \Leftrightarrow
    }
    &
    \;\;\;\;\;
    \underset{
       {Lie\;action\;property}
    }{
    \underbrace{
      \rho([x,y],z) \;=\; \rho(y,\rho(x,z)) - \rho(x,\rho(y,z))
    }
    }
    \\
    \underset{
      {\rho = -[-,-]}
      \atop
      {adjoint\;action}
    }{
      \Leftrightarrow
    }
    &
    \;\;\;\;\;
    \underset{
       {Jacobi\;identity}
    }{
    \underbrace{
      [[x,y],z] \;=\; - [y,[x,z]] + [x,[y,z]]
    }
    }
  \end{aligned}
$$

where the last line shows the equivalence to the [[Jacobi identity]] on the [[Lie algebra object]] itself in the case that the Lie action is the [[adjoint action]].

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

In the language of [[Jacobi diagrams]] this is called the _[[STU-relation]]_.
and is the reason behind the existence of [[Lie algebra weight systems]] in [[knot theory]]. For more see also at _[[metric Lie representation]]_.

### As algebras over the Lie operad

Equivalently, Lie algebra objects  are the [[algebras over an operad]] over a certain quadratic [[operad]], called the [[Lie operad]], which is the [[Koszul dual]] of the [[commutative algebra operad]].

## Examples
 {#Examples}

Examples of types of [[Lie algebra objects]]:

* If $k$ is the ring $\mathbb{Z}$ of [[integers]] and $\mathcal{C} = $ $k$[[Mod]] = [[Ab]] is the [[category]] of [[abelian groups]] equipped with the [[tensor product of abelian groups]], then a Lie algebra object is called a _[[Lie ring]]_.

* If $k$ is a [[field]] and $\mathcal{C} =$ [[Vect]] is the [[category]] of [[vector spaces]] over $k$ equipped with the [[tensor product of vector spaces]] then a Lie algebra object is an ordinary _[[Lie algebra|Lie k-algebra]]_.

* If $k$ is a [[field]] and $\mathcal{C}$ = [[sVect]] is the [[category]] of [[super vector spaces]] over $k$, then a Lie algebra object is a  _[[super Lie algebra]]_.

* A [[Leibniz algebra]] is an internal Lie algebra in the [[Loday-Pirashvili category]] ([Loday-Pirashvili 98](#LodayPirashvili98))

* [[Rozansky-Witten weight systems]] are [[Lie algebra weight systems]] for [[Lie algebra objects]] in the [[derived category of quasi-coherent sheaves]] ([Roberts-Willerton 10](#RobertsWillerton10))

## Related concepts

* [[super Lie algebra]]

* [[Lie algebra weight system]]

## References

### General

(...)

### Examples
 {#ReferencesExamples}

For references on [[super Lie algebras]] see there.

On [[Leibniz algebras]] as Lie algebra objects in suitable [[tensor categories]] of [[linear maps]]:

* {#LodayPirashvili98} [[Jean-Louis Loday]], [[Teimuraz Pirashvili]], _The tensor category of linear maps and Leibniz algebras_, Georg. Math. J. vol. 5, n.3 (1998) 263--276 ([doi:10.1023/B:GEOR.0000008125.26487.f3](https://doi.org/10.1023/B:GEOR.0000008125.26487.f3))

On [[Rozansky-Witten weight systems]] as [[Lie algebra weight systems]] for [[Lie algebra objects]] in the [[derived category of quasi-coherent sheaves]], and unified [[Wheels theorem]]:

* {#RobertsWillerton10} [[Justin Roberts]], [[Simon Willerton]], _On the Rozansky-Witten weight systems_, Algebr. Geom. Topol. 10 (2010) 1455-1519 ([arXiv:math/0602653](https://arxiv.org/abs/math/0602653))

[[!redirects Lie algebra objects]]

[[!redirects internal Lie algebra]]
[[!redirects internal Lie algebras]]

[[!redirects Lie ring]]
[[!redirects Lie rings]]


Lie algebra representation

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### Context
#### Representation theory
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[[!include representation theory - contents]]
=--
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[representation]]/[[action]] of a [[Lie algebra]] $\mathfrak{g}$ on a [[vector space]] $V$.

## Definition

A _Lie algebra representation_ is a Lie algebra [[homomorphism]] $\rho \;\colon\;\mathfrak{g} \to end(V)$ to the [[endomorphism Lie algebra]] of $V$.

This means equivalently that

$$
  \array{
    \mathfrak{g}
      \otimes
    V
    & \overset{\rho}{\longrightarrow} &
    V
    \\
    (x,v) &\mapsto& \rho_x(v)
  }
$$

is a [[bilinear map]] such that for all $x_i \in \mathfrak{g}$ and $v \in V$ we have the [[Lie action property]]:

\[
  \label{LieActionProperty}
  \rho_{x_1}(\rho_{x_2}(v))
  -
  \rho_{x_2}(\rho_{x_1}(v))
  \;=\;
  \rho_{[x_1,x_2]}(v)
\]

## Properties

### In terms of string diagrams / Jacobi diagrams
 {#InTermsOfStringDiagrams}

In [[string diagram]]-notation for [[Lie algebra objects]] [[internalization|internal]] to [[tensor categories]], the [[Lie action property]] (eq:LieActionProperty) looks as follows:

$$
  \begin{aligned}
    \Leftrightarrow
    &
    \;\;\;\;\;
    \rho(f(x,y),z) \;=\; \rho(y,\rho(x,z)) - \rho(x,\rho(y,z))
    \\
    \underset{
      {f = [-,-]}
      \atop
      {Lie\;bracket}
    }{
      \Leftrightarrow
    }
    &
    \;\;\;\;\;
    \underset{
       {Lie\;action\;property}
    }{
    \underbrace{
      \rho([x,y],z) \;=\; \rho(y,\rho(x,z)) - \rho(x,\rho(y,z))
    }
    }
    \\
    \underset{
      {\rho = -[-,-]}
      \atop
      {adjoint\;action}
    }{
      \Leftrightarrow
    }
    &
    \;\;\;\;\;
    \underset{
       {Jacobi\;identity}
    }{
    \underbrace{
      [[x,y],z] \;=\; - [y,[x,z]] + [x,[y,z]]
    }
    }
  \end{aligned}
$$

Here the last line shows the equivalence to the [[Jacobi identity]] on the [[Lie algebra object]] itself in the case that the Lie action is the [[adjoint action]].

In the language of [[Jacobi diagrams]] this is called the _[[STU-relation]]_, and is the reason behind the existence of [[Lie algebra weight systems]] in [[knot theory]].

## Related concepts

* [[embedding tensor]]

* [[Casimir invariant]]

* [[Lie algebra cohomology]]

* [[group action]]

* [[Lie infinity-algebroid representation]]

## References

See also

* Wikipedia, _[Lie algebra representation](https://en.wikipedia.org/wiki/Lie_algebra_representation)_

[[!redirects Lie algebra representations]]

[[!redirects Lie algebra action]]
[[!redirects Lie algebra actions]]

[[!redirects Lie module]]
[[!redirects Lie modules]]

[[!redirects Lie action]]
[[!redirects Lie actions]]
[[!redirects Lie action property]]
[[!redirects Lie action properties]]

[[!redirects Lie algebra module]]
[[!redirects Lie algebra modules]]


Lie algebra weight system

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### Context
#### Knot theory
+-- {: .hide}
[[!include knot theory - contents]]
=--
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In [[knot theory]], a large class of [[weight systems]] arises from reading a ([[horizontal chord diagram|horizontal]]) [[chord diagram]] as a [[string diagram]] in the evident way, and then labeling it by the structure morphisms of a [[metric Lie algebra|metric]] [[Lie algebra object]] equipped with a [[metric Lie representation|metric]] [[Lie algebra representation]] [[internalization|internal to]] a suitable [[tensor category]].

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

This does yield [[weight systems]], because -- using that [[chord diagrams modulo 4T are Jacobi diagrams modulo STU]] -- the required [[STU-relations]] translate into the structural equations satisfied by [[Lie modules]] ([[Jacobi identity]] and [[Lie action property]]); see [Roberts & Willerton 06, Theorem 3.1](#RobertsWillerton06), following [Vaintrob 94](#Vaintrob94), [Vogel 11](#Vogel11) following [Bar-Natan 95, 2.4](#BarNatan95), [Bar-Natan 96](#BarNatan96) following [Kontsevich 94](#Kontsevich94).

The [[weight systems]] arising this way are called _Lie algebra weight systems_.

## Properties

### Ubiquity

#### On round chord diagrams

Examples of [[weight systems]] on (ordinary, round) [[chord diagrams]] which are _not_ Lie algebra weight systems are rare. Originally it was conjectured that none exist ([Bar-Natan 95, Conjecture 1](#BarNatan95), [Bar-Natan & Stoimenow 97, Conjecture 2.4](#BarNatanStoimenow97)).

Eventually, a (counter-)example of a weight system which at least does not arise from any [[finite dimensional vector space|finite-dimensional]] [[super Lie algebra]] was given in [Vogel 11](#Vogel11).

#### On horizontal chord diagrams

For [[horizontal chord diagrams]] we have:

* _[[all horizontal weight systems are partitioned Lie algebra weight systems]]_ ([Bar-Natan 96](#BarNatan96))

* the [[fundamental representation|fundamental]] [[general linear Lie algebra|$\mathfrak{gl}(n)$]]-weight system coincides with the [[Cayley distance kernel]] at [[inverse temperature]] $\beta = \ln(n)$ ([CSS21](#CSS21))

### Stringy weight systems span classical Lie algebra weight systems

[[stringy weight systems span classical Lie algebra weight systems]]

### Relation to Adams operations

On the [[vector space]] $\mathcal{A}$ of [[Jacobi diagrams]] [[quotient vector space|modulo]] [[STU-relations]] ([[chord diagrams modulo 4T are Jacobi diagrams modulo STU|equivalently]] [[chord diagrams]] [[quotient vector space|modulo]] [[4T relations]])  there is a system of [[linear maps]] (for $q \in \mathbb{Z}$, $q \neq 0$)

$$
  \psi^q
  \;\colon\;
  \mathcal{A}
  \longrightarrow
  \mathcal{A}
$$

which respect the [[coalgebra]] [[mathematical structure|structure]] and satisfy

$$
  \psi^{q_2} \circ \psi^{q_1}
  \;=\;
  \psi^{q_1 \cdot q_2}
$$

and as such are (dually) analogous to the [[Adams operations]] on [[topological K-theory]].

([Bar-Natan 95, Def. 3.11 & Theorem 7](#BarNatan95))

In fact, when evaluated in [[Lie algebra weight systems]] $w_{\mathbf{N}}$ and under the identification (see [here](equivariant+K-theory#RelationToRepresentationTheory)) of the [[representation ring]] of a [[compact Lie group]] $G$ with the $G$-[[equivariant K-theory]] of the point, these Adams operations on [[Jacobi diagrams]] correspond to the [[Adams operations]] on [[equivariant K-theory]]:

$$
  w_{\mathbf{N}}(\psi^q D)
  \;=\;
  w_{\psi^q \mathbf{N}}(D)
  \,.
$$

([Bar-Natan 95, Exc. 6.24](#BarNatan95)
)

For more see at _[[Adams operation on Jacobi diagrams]]_.

### As expectation values of single trace observables
 {#AsExpectationValuesOfSingleTraceOperators}

We discuss how [[Lie algebra weight systems]] on round [[chord diagrams]] arise in [[quantum field theory]] and [[statistical mechanics]] as [[expectation values]] of suitable [[single trace observables]] subject to [[Wick's theorem]].

So consider

* $\mathfrak{g}$ be a [[metric Lie algebra]], with metric denoted $k$,

* $(\mathfrak{g}\otimes C \overset{\rho}{\to} C) \in \mathfrak{g}Mod$ be a [[finite-dimensional vector space|finite-dimensional]] [[Lie algebra representation]] of $\mathfrak{g}$.

For comparison with traditional literature, choose

1. a [[linear basis]] $\{v^a\}$ of $C$

1. a [[linear basis]] $\{t^i\}$ of $\mathfrak{g}$

In terms of these linear bases, the representation is given by a sequence $\rho^i$ of [[square matrices]] $\big((\rho^i)^a{}_b\big)$ such that

$$
  (\rho^i)^a{}_b v^b
  \;\coloneqq\;
  \rho(t^i, v^a)
$$

with the [[Einstein summation convention]] understood here and in the following.

We write $k = (k_{i j})$ for the components of the Lie algebra metric in this basis, and write

$$
  \rho_i \;\coloneqq\; k_{i j} \rho^j
$$

With all this understood, a [[field (physics)|field]]/[[random variable]] with values in $\mathfrak{g}$ is

$$
  Z \;=\; Z_i \cdot \rho^i \;\;
$$

for [[scalar field]] components $Z_i$.

Now assume that the $Z_i$ are [[free fields]]/[[random variables]] with [[Gaussian distribution]], hence such that

$$
  \langle Z_i \rangle \;=\; 0
$$

$$
  \langle Z_i Z_j \rangle \;=\; k_{i j}
$$

and such that [[Wick's theorem]] applies to higher [[moments]]:

$$
  \langle Z_{i_1} Z_{i_2} Z_{i_3} Z_{i_4} \rangle
  \;=\;
  k_{i_1 i_2} k_{i_3 i_4}
  +
  k_{i_1 i_3} k_{i_2 i_4}
  +
  k_{i_1 i_4} k_{i_2 i_3}
$$

etc.

Consider the [[expectation values]] of [[single trace observables]]:

$$
  \Big\langle
    tr
    \big(
      \underset{
        \mathclap{
          n\, factors
        }
      }{
      \underbrace{
        Z Z \cdots Z
      }
      }
    \big)
  \Big\rangle
  \;\coloneqq\;
  \big\langle
    Z_{i_1}
    Z_{i_2}
    \cdots
    Z_{i_n}
  \big\rangle
  \cdot
  tr
  \big(
    \rho^{i_1}
    \cdot
    \rho^{i_2}
    \cdots
    \rho^{i_n}
  \big)
$$

Using [[Wick's theorem]], they are given by

$$
  \begin{aligned}
    \Big\langle
      tr
      \big(
        Z Z Z Z
      \big)
    \Big\rangle
    &
    =
  \big\langle
    Z_{i_1}
    Z_{i_2}
    Z_{i_3}
    Z_{i_4}
  \big\rangle
  \cdot
  tr
  \big(
    \rho^{i_1}
    \cdot
    \rho^{i_2}
    \cdot
    \rho^{i_3}
    \cdot
    \rho^{i_4}
  \big)
  \\
  & =
  \phantom{+\;}
  k_{i_1 i_2} \, k_{i_3 i_4}
  \cdot
  tr
  \big(
    \rho^{i_1}
    \cdot
    \rho^{i_2}
    \cdot
    \rho^{i_3}
    \cdot
    \rho^{i_4}
  \big)
  \\
  &
  \phantom{=\;}
  +
  k_{i_1 i_3} \, k_{i_2 i_4}
  \cdot
  tr
  \big(
    \rho^{i_1}
    \cdot
    \rho^{i_2}
    \cdot
    \rho^{i_3}
    \cdot
    \rho^{i_4}
  \big)
  \\
  &
  \phantom{=\;}
  +
  k_{i_1 i_4} \, k_{i_2 i_3}
  \cdot
  tr
  \big(
    \rho^{i_1}
    \cdot
    \rho^{i_2}
    \cdot
    \rho^{i_3}
    \cdot
    \rho^{i_4}
  \big)
  \\
  & =
  2
  \cdot
  tr
  \big(
    \rho_i
    \cdot
    \rho^i
    \cdot
    \rho_j
    \cdot
    \rho^j
  \big)
  \\
  &
  \phantom{=\;}
  +
  tr
  \big(
    \rho_{i}
    \cdot
    \rho_j
    \cdot
    \rho^i
    \cdot
    \rho^j
  \big)
  \end{aligned}
$$

etc.

{#TracedWickTheorem} Here [[Wick's theorem]] in the first lines is given by a sum over [[linear chord diagrams]], and the [[trace]] then serves to close these to [[round chord diagrams]]:

> from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

The values of the traces in the last line are
the values of the [[Lie algebra weight system]] $w_C$ on the [[chord diagram]] which reflects the Wick-contractions modulo cyclic permutation.

Essentially this observation (specifically for [[SYK-model]]-like systems and without mentioning of [[weight systems]]) appears in [GGJJV 18, Section 2.2](#GGJJV18), [Jia-Verbaarschot 18, Section 4](#JiaVerbaarschot18), [BNS 18, Section 2.1](#BNS18), [BINT 18, Section 2](#BINT18), [Narovlansky 19, Slides 5-21](#Narovlansky19).

[[!include chord diagrams as multi-trace observables in BMN matrix model -- example]]

## Related concepts

* [[metric Lie algebra]]

* [['t Hooft double line notation]]

[[!include chord diagrams and weight systems -- table]]

## References

### General

Lie algebra weight systems motivated from [[Yang-Mills theory]] [[Feynman amplitudes]]:

* [[Predrag Cvitanović]], _Group theory for Feynman diagrams in non-Abelian gauge theories_, Phys. Rev. D14 (1976) 1536-1553 ([doi:10.1103/PhysRevD.14.1536](https://doi.org/10.1103/PhysRevD.14.1536), [spire:108133](http://inspirehep.net/record/108133), [[CvitanovicWeights76.pdf:file]])

The concept of round Lie algebra weight systems for [[Chern-Simons theory]] with [[Wilson lines]] originates around

* {#Kontsevich94} [[Maxim Kontsevich]], _Feynman diagrams and low-dimensional topology_, First European Congress of Mathematics, 1992, Paris, vol. II, Progress in Mathematics __120__, Birkhäuser (1994), 97–121 ([pdf](http://www.ihes.fr/~maxim/TEXTS/Feynman%20%20diagrams%20and%20low-dimensional%20topology.pdf))

* {#BarNatan95} [[Dror Bar-Natan]], Section 2.4 of: _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))

and for [[horizontal weight systems]] in

* {#BarNatan96} [[Dror Bar-Natan]], _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001))

reviewed in

* {#BarNatanStoimenow97} [[Dror Bar-Natan]], Alexander Stoimenow, Section 2.2 of: _The Fundamental Theorem of Vassiliev Invariants_, in: *Geometry and Physics*, Lecture Notes in Pure and Applied Mathematics, volume 184, Marcel Dekker Inc. 1996 ([arXiv:q-alg/9702009](https://arxiv.org/abs/q-alg/9702009), [ISBN:0-8247-9791-4](https://users-math.au.dk/swann/Aarhus-proceedings-1995/index.html))

Discussion of [[fundamental representation|fundamental]] $\mathfrak{gl}(n)$-weight systems in terms of [[Cayley distance kernels]] on the [[symmetric group]]:

* {#CSS21} [[David Corfield]], [[Hisham Sati]], [[Urs Schreiber]]: *[[schreiber:Fundamental weight systems are quantum states]]* ([arXiv:2105.02871](https://arxiv.org/abs/2105.02871))

### Via string diagrams

From the construction given in [Bar-Natan 95, Section 2.4](#BarNatan95) the interpretation of [[Lie algebra weight systems]] in terms of [[string diagrams]] for [[Lie algebra objects]] in [[tensor categories]] is evident, but standard textbooks in [[knot theory]]/[[combinatorics]] do not pick this up:

* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], Chapter 6 of: _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv:1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))

* {#JacksonMoffat19} [[David Jackson]], [[Iain Moffat]], Section 14 of: _An Introduction to Quantum and Vassiliev Knot Invariants_, Springer 2019 ([doi:10.1007/978-3-030-05213-3](https://link.springer.com/book/10.1007/978-3-030-05213-3))

The interpretation of Lie algebra weight systems as [[string diagram]]-calculus and generalization to [[Lie algebra objects]] (motivated by generalization at least to [[super Lie algebras]]) is made more explicit in

* {#Vaintrob94} [[Arkady Vaintrob]], _Vassiliev knot invariants and Lie S-algebras_, Mathematical Research Letters1, 579–595 (1994) ([pdf](https://pdfs.semanticscholar.org/bdc3/ac1d8da476245e2408e481a70b115b3e9aab.pdf))

* {#Vogel11} [[Pierre Vogel]], _Algebraic structures on modules of diagrams_, Journal of Pure and Applied Algebra, Volume 215, Issue 6, June 2011, Pages 1292-1339 ([doi:10.1016/j.jpaa.2010.08.013](https://doi.org/10.1016/j.jpaa.2010.08.013), [pdf](https://webusers.imj-prg.fr/~pierre.vogel/diagrams.pdf))

and fully explicit in

* {#RobertsWillerton06} [[Justin Roberts]], [[Simon Willerton]], Section 3 of: _On the Rozansky-Witten weight systems_, Algebr. Geom. Topol. 10 (2010) 1455-1519 ([arXiv:math/0602653](https://arxiv.org/abs/math/0602653))

See also

* [[Vladimir Hinich]], [[Arkady Vaintrob]], _Cyclic operads and algebra of chord diagrams_, Sel. math., New ser. (2002) 8: 237 ([arXiv:math/0005197](https://arxiv.org/abs/math/0005197))

* Alexander Schrijver, _On Lie algebra weight systems for 3-graphs_ ([arXiv:1412.6923](https://arxiv.org/abs/1412.6923))

### For $\mathfrak{sl}(2)$

For the [[special linear Lie algebra]] $\mathfrak{sl}(2)$

* [[Sergei Chmutov]], [[Alexander Varchenko]], _Remarks on the Vassiliev knot invariants coming from $\mathfrak{sl}(2)$_, Topology 36 (1), 153-178, 1997 (doi:10.1016/0040-9383(95)00071-2)

* S. Tyurina, [[Alexander Varchenko]], *A remark on $\mathfrak{sl}_2$ approximation of the Kontsevich integral of the unknot*, Journal of Mathematical Sciences **131** (2005) 5270–5274  ([arXiv:math/0111201](https://arxiv.org/abs/math/0111201), [doi:10.1007/s10958-005-0399-1](https://doi.org/10.1007/s10958-005-0399-1))

* E. Kulakova, S. Lando, T. Mukhutdinova, G. Rybnikov, _On a weight system conjecturally related to $\mathfrak{sl}_2$_, European Journal of Combinatorics Volume 41, October 2014, Pages 266-277 ([arXiv:1307.4933](https://arxiv.org/abs/1307.4933))

* [Jackson & Moffat 19, Sec. 14.4](#JacksonMoffat19)

### For $\mathfrak{sl}(N)$

For the [[special linear Lie algebra]] $\mathfrak{sl}(N)$ for all $N$:

* [[Fumikazu Nagasato]], *A Diagrammatic Construction of the $(sl(N,\mathbb{C}, \rho))$-Weight System*, Interdisciplinary Information Sciences, **9** 1 (2003) 43-51 ([doi:10.4036/iis.2003.43](https://doi.org/10.4036/iis.2003.43))

* [[Fumikazu Nagasato]], *An Exposition of the $\mathfrak{sl}(N,\mathbb{N})$-Weight System*, Journal of Lie Theory 17 (2007), No. 2, 263--281, Heldermann Verlag 2007 ([jlt:17014](https://www.heldermann.de/JLT/JLT17/JLT172/jlt17014.htm))

### For $\mathfrak{gl}(1\vert1)$

For the [[super Lie algebra]] $\mathfrak{gl}(1\vert1)$:

* {#FFKV97} [[José Figueroa-O’Farrill]], [[Takashi Kimura]], [[Arkady Vaintrob]], _The universal Vassiliev invariant for the Lie superalgebra $\mathfrak{gl}(1\vert1)$_, Commun. Math. Phys. 185 (1997) 93-127 ([arXiv:q-alg/9602014](https://arxiv.org/abs/q-alg/9602014))

### Further discussion

Relation of the [[colored Jones polynomial]] to [[Lie algebra weight systems]] on [[chord diagrams]]:

* [[Dror Bar-Natan]], [[Stavros Garoufalidis]], _On the Melvin–Morton–Rozansky conjecture_, Invent math (1996) 125: 103 ([doi:10.1007/s002220050070](https://doi.org/10.1007/s002220050070), [[BarNatanGaroufaldis96.pdf:file]])

On the [[logical equivalence]] between the [[four-colour theorem]] and a statement about transition from the [[small N limit]] to the [[large N limit]] for [[Lie algebra weight systems]] on [[Jacobi diagrams]] via the [['t Hooft double line construction]]:

* [[Dror Bar-Natan]], _Lie Algebras and the Four Color Theorem_, Combinatorica 17-1(1997) 43–52  ([arXiv:q-alg/9606016](https://arxiv.org/abs/q-alg/9606016), [doi:10.1007/BF01196130](https://doi.org/10.1007/BF01196130))

[[!include weight systems on chord diagrams in physics]]

[[!redirects Lie algebra weight systems]]


Lie algebras > history

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


Lie algebroid

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

#Contents#
* automatic table of contents goes here
{:toc}

## Definition ##

A _Lie algebroid_ is the [[horizontal categorification|many object version]] of a [[Lie algebra]]. It is the [[infinitesimal space|infinitesimal]] approximation to a [[Lie groupoid]].

There are various equivalent definitions:

### In terms of vector bundles with anchor ###

+-- {: .num_defn}
###### Definition in terms of vector bundles with anchor map

A Lie algebroid over a [[manifold]] $X$ is

 * a [[vector bundle]] $E \to X$;

 * equipped with a Lie brackets $[\cdot,\cdot] : \Gamma(E)\otimes \Gamma(E) \to \Gamma(E)$ (over the ground field) on its space of sections;

 * a morphisms of vector bundles $\rho : E \to TX$, whose tangent map preserves the bracket: $(d\rho)([\xi,\zeta]_{\Gamma E}) = [d\rho(\xi),d\rho(\zeta)]_{\Gamma TX}$; (but this property of preserving brackets is implied by the next property, see [[Yvette Kosmann-Schwarzbach]], F. Magri, _Poisson-Nijenhuis structures_, Ann. Inst.
H. Poincaré Phys. Théor., 53(1):3581, 1990.)

 * such that the _Leibniz rule_ holds: for all $X, Y \in \Gamma(E)$ and all $f \in C^\infty(X)$ we have
$$
  [X, f \cdot Y] = f\cdot [X,Y] + \rho(X)(f) \cdot Y
  \,.
$$
=--

### The CE-algebra of a vector bundle with anchor ###

Given this data of a vector bundle $E \to X$ with anchor map $\rho$ as above, one obtains the structure of a [[dg-algebra]] on the [[exterior algebra]] $\wedge^\bullet_{C^\infty(X)} \Gamma(E)^*$ of smooth sections of the dual bundle by the formula

$$
  (d\omega)(e_0, \cdots, e_n)
  =
  \sum_{\sigma \in Shuff(1,n)} sgn(\sigma) \rho(e_{\sigma(0)})(\omega(e_{\sigma(1)}, \cdots, e_{\sigma(n)}))
  +
  \sum_{\sigma \in Shuff(2,n-1)} sign(\sigma)
   \omega([e_{\sigma(0)},e_{\sigma(1)}],e_{\sigma(2)}, \cdots, e_{\sigma(n)} )
  \,,
$$

for all $\omega \in \wedge^n_{C^\infty(X)} \Gamma(E)^*$ and $(e_i \in \Gamma(E))$, where $Shuff(p,q)$ denotes the set of $(p,q)$-[[shuffle]]s $\sigma$ and $sgn(\sigma)$ the [[signature]] $\in \{\pm 1\}$ of the corresponding [[permutation]].

More details on this are at [[Chevalley-Eilenberg algebra]].

Conversely, one finds that every [[semi-free dga]] finitely generated in degree 1 over $C^\infty(X)$ arises this way, so that one may turn this around:

### Semi-free dg-algebras ###

+-- {: .num_defn}
###### Definition in terms of Chevalley--Eilenberg algebra

A Lie algebroid over a manifold $X$ is a vector bundle
$E \to X$ equipped with a degree +1 derivation $d$ on the free (over $C^\infty(X)$) graded-commutative algebra $\wedge^\bullet_{C^\infty(X)} \Gamma(E)^*$ (where the dual is over $C^\infty$), such that $d^2 = 0$.
=--

This is for $\Gamma(E)$ satisfying suitable finiteness conditions.  In general, as the masters well knew, the correct definition is the algebra of alternating multilinear functions from  $\Gamma(E)$ to the ground field, assumed of characteristic 0. This can also be phrased in terms of
linear maps from the corresponding coalgebra cogenerated by
$\Gamma(E)$, but the masters did not have coalgebras in those days.

The differential graded-commutative algebra

$$
  CE(\mathfrak{g})
  :=
  (\wedge^\bullet_{C^\infty(X)} \Gamma(E)^*, d)
$$

is the [[Chevalley-Eilenberg algebra]] of the Lie algebroid (in that for $X = pt$ it reduces to the ordinary Chevally--Eilenberg algebra for Lie algebras).

In the existing literature this is often addressed just as "the complex that computes Lie algebroid cohomology".

It is helpful to compare this definition to the general definition of [[Lie infinity-algebroid|Lie ∞-algebroids]], the [[vertical categorification]] of Lie algebras and Lie algebroids.

### Lie-Rinehart algebras ###

+-- {: .num_defn}
###### Definition in terms of commutative Lie--Rinehart pairs

A Lie algebroid over the manifold $X$ is

 * a Lie algebra $\mathfrak{g}$;

 * the structure of a Lie module over $\mathfrak{g}$ on $C^\infty(X)$ (i.e. an action of $\mathfrak{g}$ on $X$);

 * the structure of a $C^\infty(X)$-module on $\mathfrak{g}$ (in fact: such that $\mathfrak{g}$ is a finitely generated projective module);

 * such that the two actions satisfy two compatibility conditions which are modeled on the standard relations obtained by setting $\mathfrak{g} = \Gamma(T X)$.
=--

This is the special case of a [[Lie-Rinehart pair]] $(A,\mathfrak{g})$ where the associative algebra $A$ is of the form $C^\infty(X)$.

## Examples ##

* A **[[Lie algebra]]** is a Lie algebroid over a point, $X = pt$.

* The **[[tangent Lie algebroid]]** is

    1. in the vector bundle definition given by $E = T X$, $\rho = \mathrm{Id}$;

    2. in the [[Chevalley-Eilenberg algebra]] definition: $\mathrm{CE}(T X) = (\Omega^\bullet(X), d_{deRham})$;

* An [[action Lie algebroid]] is the Lie version of an [[action groupoid]].

*  **Bundles of Lie algebras** $E \to X$ with fiber $\mathfrak{g}$ are Lie algebroids with $\rho = 0$ and fiberwise bracket. In particular, for $G$ a Lie group with Lie algebra $\mathfrak{g}$ and $P \to X$ a $G$-principal bundle, the _adjoint bundle_ $ad P := P \times_G \mathfrak{g}$ (where $\mathfrak{g}$ is associated using the [[adjoint representation]] of $G$ on its Lie algebra) is a bundle of Lie algebras.

* Lie algebroids with injective anchor maps are equivalently [[integrable distributions]] in the [[tangent bundle]] of their base manifold and hence are equivalently [[foliations]] of their base manifold.

* The **[[Atiyah Lie algebroid]]** is the Lie algebroid of the  [[Atiyah Lie groupoid]] of a principal bundle: for $G$ a Lie group and $P \to X$ a $G$-principal bundle, the vector bundle $At(P):= T P/G$ naturally inherits the structure of a Lie algebroid. Moreover, it fits into a short exact sequence of Lie algebroids over $X$
$$
  0 \to ad P \to At(P) \to T X \to 0
$$
known as the **Atiyah sequence**.

* The **vertical tangent Lie algebroid** $T_{vert}Y \hookrightarrow T Y$ of a smooth map $\pi : Y \to X$ of manifolds is the sub-Lie algebroid of the tangent Lie algebroid $T Y$ defined as follows:

   1. In the vector bundle perspective $E = ker(\pi_*)$ is the kernel bundle of the map $\pi_* : T Y \to T X$.

   2. In the dual picture we have $CE(T_{vert}Y) = \Omega^\bullet_{vert}(Y)$, the [[Lie infinity-algebroid|qDGCA]] of **vertical differential forms**. This is the quotient of $\Omega^\bullet(Y)$ by the ideal of those forms which vanish when restricted in all arguments to $ker(\pi_*)$.

* Each [[Poisson manifold]] $(X,\pi)$ defines and is defined by a [[Poisson Lie algebroid]] $T^* X \stackrel{\pi}{\to} t X$. This is the degree-1 example of a more general structure described at [[n-symplectic manifold]].

* If $E\to X$ is a Lie algebroid with bracket $[,]$ and anchor $\rho:E\to TX$ then it induces a Lie algebroid structure on the $k$-th [[jet bundle|jet]] bundle $j^k E\to X$, called the **jet Lie algebroid**. More precisely, if $s\in\Gamma_X E$ then call by $j^k s$ the induced section in $\Gamma_X j^k E$. Then there is a unique Lie algebroid structure on the bundle $j^k E\to X$ such that the following two properties hold: $[j^k s, j^k t] = j^k [s,t]$ and $\rho(j^k s) = \rho(s)$ for all $s,t\in\Gamma_X E$ (see [pdf](http://www.math.ist.utl.pt/~rfern/Meus-papers/HTML/jets.pdf)).

* A [[BRST complex]] is a [[Chevalley-Eilenberg algebra]] of a Lie algebroid which corresponds to the [[action groupoid]] of a Lie group acting on a space.

## Properties

### Lie theory

The extent to which Lie algebroids are to [[Lie groupoid|Lie groupoids]] as Lie algebras are to Lie groups is the content of general [[Lie theory]], in which [[Lie's three theorems|Lie's theorems]] have been generalized to Lie algebroids.

### Poisson geometry

The fiberwiese linear dual of a Lie algebroid (regarded as a vector bundle) is naturally a [[Poisson manifold]]: the _[[Lie-Poisson structure]]_.

## Related concepts

* [[Lie algebra]], **Lie algebroid**

* [[double Lie algebroid]]

* [[L-∞ algebra]], [[L-∞ algebroid]]

* [[foliation of a Lie algebroid]]

* [[Lie algebroid cohomology]]

* [[super Lie algebroid]]

[[!include oidification - table]]

## References

The concept of Lie algebroid was introduced in

* [[Jean Pradines]], _Théorie de Lie pour les groupoïdes différentiables. Calcul différenetiel dans la catégorie des groupoïdes infinitésimaux_, C. R. Acad. Sci. Paris Sér. A-B 264 1967 A245--A248, [MR0216409](http://www.ams.org/mathscinet-getitem?mr=0216409)

In algebra a generalization of Lie algebroid, the Lie pseduoalgebra or Lie-Rinehart algebra/pair has been introduced more than a dozen of times under various names starting in early 1950-s. Atiyah's construction of Atiyah sequence is published in 1957 and Rinehart's paper in 1963.

Historically important is also the reference

* [[Theodore Courant]], _Tangent Lie algebroid_ ([pdf](http://www.iop.org/EJ/article/0305-4470/27/13/026/ja941326.pdf))

on [[tangent Lie algebroids]].

A bijective correspondence between Lie algebroid structures, homological vector fields of degree 1, and odd linear Poisson structures is established in the paper

* [[A. Yu. Vaintrob]], _Lie algebroids and homological vector fields_, Russian Mathematical Surveys 52:2 (1997), 428–429.  [doi](http://dx.doi.org/10.1070/rm1997v052n02abeh001802), [[vaintrob.pdf:file]].

Textbook accounts

* [[Kirill Mackenzie]], _General theory of Lie groupoids and Lie algebroids,_ Cambridge University Press, 2005, xxxviii + 501 pages  ([website](http://kchmackenzie.staff.shef.ac.uk/gt.html))

* [[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 ([MathSciNet](http://www.ams.org/mathscinet-getitem?mr=896907))

* [[Janez Mrčun]], [[Ieke Moerdijk]], _Introduction to foliations and Lie groupoids_, Cambridge Studies in Advanced Mathematics __91__, Cambridge University Press 2003. x+173 pp. ISBN: 0-521-83197-0

Expository review (motivated by [[higher spin gauge theory]]):

* [[Xavier Bekaert]], *Geometric tool kit for higher spin gravity (part II): An introduction to Lie algebroids and their enveloping algebras* [[arXiv:2308.00724](https://arxiv.org/abs/2308.00724)]

For an infinite-dimensional version used in stochastic analysis see

* Rémi Léandre, _A Lie algebroid on the Wiener space_, Adv. Math. Phys. 2010, Art. ID 146719, 17 pp. [MR2011j:58064](http://www.ams.org/mathscinet-getitem?mr=2594930)

There is also a recent "hom-version"

* Camille Laurent-Gengoux, Joana Teles, _Hom-Lie algebroids_, [arxiv/1211.2263](http://arxiv.org/abs/1211.2263)

[[!redirects Lie algebroids]]

[[!redirects anchor map]]
[[!redirects anchor maps]]


Lie algebroid-groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Internal categories
+--{: .hide}
[[!include internal infinity-categories contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[internal groupoid]] in the [[category]] of [[Lie algebroids]].

## Examples

For $\mathcal{G}_\bullet$ a [[Lie groupoid]], forming degreewise the [[tangent Lie algebroid]] yields the tangent Lie algebroid groupoid

$$
  \array{
    T \mathcal{G}_1 &\to& \mathcal{G}_1
    \\
    {}^{\mathllap{d s}}\downarrow \downarrow^{\mathrlap{d t}}
    &&
    {}^{\mathllap{s}}\downarrow \downarrow^{\mathrlap{t}}
    \\
    T \mathcal{G}_0 &\to& \mathcal{G}_0
  }
  \,.
$$

## Related concepts

* [[double Lie algebroid]], **Lie algebroid-groupoid**, [[double Lie groupoid]]

* [[foliation of a Lie groupoid]]

## References

The notion is considered (under the name "$\mathcal{L A}$-groupoids") in

* [[Kirill Mackenzie]], _Double Lie algebroids and second-order geometry. I_. Adv. Math. 94 (1992), no. 2, 180–239

* [[Kirill Mackenzie]], _Double Lie algebroids and second-order geometry. II_. Adv.Math. 154 (2000), no. 1, 46–75. ([arXiv:dg-ga/9712013](http://arxiv.org/abs/dg-ga/9712013)).

[[!redirects Lie algebroid-groupoids]]


Lie bialgebra

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

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Quantization
 {#Quantization}

### Idea

[[quantum group|Quantum groups]] were introduced independently by Drinfeld and Jimbo around 1984. One of the most important examples of quantum groups are deformations of
[[universal enveloping algebras]]. These [[deformations]] are closely related to Lie bialgebras. In particular, every deformation of a universal enveloping algebra induces
a Lie bialgebra structure on the underling Lie algebra. In ([Drinfeld](#Drinfeld)) Drinfeld asked if the converse of this statement holds:

"Does there exist a universal quantization for Lie bialgebras?"

This was answered to the positive in ([Etingof-Kazhdan](#EtingofKazhdan)).

## References

* [[Vladimir Drinfeld]], _On some unsolved problems in quantum group theory_,  Lecture Notes in Math., 1510, Springer, Berlin, 1992.
 {#Drinfeld}

* P. Etingof , D. Kazhdan, _Quantization of Lie bialgebras_ , I, Selecta Math. 2 (1996) n. 1, 1-41.
 {#EtingofKazhdan}

* Wikipedia, _[Lie biablgebra](http://en.wikipedia.org/wiki/Lie_bialgebra)_
* Štefan Sakáloš, [[Pavol Ševera]], _On quantization of quasi-Lie bialgebras_, [arxiv/1304.6382](http://arxiv.org/abs/1304.6382)

[[!redirects Lie bialgebras]]


Lie bialgebroid

#Idea#

(...something like "[[Lie algebroid]] internal to Lie algebroids", but subtle...)

#Definition#

Given a [[Lie algebroid]] $A$ together with the structure of a [[Lie algebroid]] $A^*$ on the dual of the vector bundle underlying $A$, the interpretation of a Lie algebroids as a [[supermanifold]] as described at [[Lie infinity-algebroid]] induces two notions of differentials $d_A$ and $d_{A^*}$ and two notions of Schouten brackets.

A pairs $(A,A^*)$ of Lie algebroids is a **Lie bialgebroid** if these differentials are derivations of the corresponding Schouten brackets.

See for instance [definition 2.2.2](http://arxiv.org/PS_cache/math/pdf/9910/9910078v1.pdf#page=15) in _Roytenberg99_.

#Remarks#

* Every Lie bialgebroid $(A,A*)$ induces the structure of a [[Courant algebroid]] on $E := A \oplus A^*$. This is [theorem 2.3.3](http://arxiv.org/PS_cache/math/pdf/9910/9910078v1.pdf#page=18) in _Roytenberg99_.
For $A = T X$ the tangent Lie algebroid of a manifold $X$, this is the crucial fact underlying [[generalized complex geometry]].

#References#

* **Roytenberg99** Dmitry Roytenberg, _Courant algebroids, derived brackets and even symplectic supermanifolds_ ([arXiv](http://arxiv.org/abs/math.DG/9910078))


Lie bracket of vector fields

+-- {: .rightHandSide}
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###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Given a [[smooth manifold]] $X$, the __Lie bracket__ of [[vector fields]] $u$ and $v$ can be defined in several ways.

### As commutator of derivations

Since [[derivations of smooth functions are vector fields]], we can identify $u$ and $v$ with the corresponding derivations $C^\infty(X)\to C^\infty(X)$.

Taking the [[commutator]] $u v-v u$ of these derivations produces another [[derivation]], which is denoted by $[u,v]$, and which can be identified with a vector field on $X$.

### As a Lie derivative

Alternatively, we can set
$$[u,v]=\mathcal{L}_u v=-\mathcal{L}_v u,$$
where $\mathcal{L}$ denotes the [[Lie derivative]] of a [[vector field]].

## Properties

The real [[vector space]] of [[vector fields]] on $X$ equipped with the Lie bracket forms a [[Lie algebra]].

## Related concepts

* [[Schouten bracket]]

* [[vector field]]

* [[Lie derivative]]

* [[Lie algebra]]


Lie category

## Definition

An [[internal category]] object in the category of [[smooth manifolds]] in which the source and target maps are [[submersions]].

Sometimes, the smooth manifold of morphisms is allowed to have a boundary, in which case the restrictions of the source and target maps to the boundary are required to be [[submersions]] themselves.

## Related concepts

* [[Lie groupoid]], [[internal category]]

## References

The notion of Lie category 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))

which also allowed for $C^r$ manifolds and structure maps.

A more recent study, including the case involving manifolds with boundary, is

* Žan Grad, _Fundamentals of Lie categories_, [arXiv:2302.05233](https://arxiv.org/abs/2302.05233).


Lie coalgebra

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

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[Lie algebra]]

* [[coalgebra]]

* [[dg-coalgebra]]

## References

On [[Lie coalgebras]], [[rational homotopy theory]] and [[Hopf invariants]] via [[homotopy Whitehead integrals]] ("[[functional cup products]]"):

* {#SinhaWalter13} [[Dev Sinha]], [[Ben Walter]], _Lie coalgebras and rational homotopy theory II: Hopf invariants_, Trans. Amer. Math. Soc. 365 (2013), 861-883  ([arXiv:0809.5084](https://arxiv.org/abs/0809.5084), [doi:10.1090/S0002-9947-2012-05654-6](https://doi.org/10.1090/S0002-9947-2012-05654-6))

See also

* Wikipedia, _[Lie coalgebra](https://en.wikipedia.org/wiki/Lie_coalgebra)_

[[!redirects Lie coalgebras]]


Lie derivative

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[smooth manifold]] $M$ and a vector field $X \in \Gamma(T M)$ on it, one defines a series of operators $\mathcal{L}_X$ on spaces of differential forms, of functions, of vector fields and multivector fields.
For functions $\mathcal{L}_X(f) = X(f)$ (derivative of $f$ along an integral curve of $X$); as multivector fields and forms can not be compared in different points, one pullbacks or pushforwards them to be able to take a derivative.

For vector fields $\mathcal{L}_X Y = [X,Y]$. If $\omega \in \Omega^\bullet(M)$ is a [[differential form]] on $M$, the **Lie derivative** $\mathcal{L}_X \omega$ of $\omega$ along $X$ is the linearization of the pullback of $\omega$ along the flow $\exp(X -) : \mathbb{R} \times M\to M$ induced by $X$

$$
  \mathcal{L}_X \omega = \frac{d}{d t}|_{t = 0} \exp(t X)^*(\omega)
  \,.
$$
Denote by $\iota_X : \Omega^\bullet(M) \to \Omega^{\bullet -1}(M)$ be the graded [[derivation]] which is the [[contraction]] with a vector field $X$. By [[Cartan's homotopy formula]],

$$
  \mathcal{L}_v = [d_{dR}, \iota_v] = d_{dR} \circ \iota_v + \iota_v \circ d_{dR} : \Omega^\bullet(X) \to \Omega^\bullet(X)
  \,.
$$

## References

Cartan introduced Lie derivatives of differential forms and derived [[Cartan's magic formula]] in

* [[Élie Cartan]], _Leçons sur les invariants intégraux_ (based on lectures given in 1920-21 in Paris, Hermann, Paris 1922, reprinted in 1958).

Extension to arbitrary tensor fields was given in

* W. Ślebodziński, Sur les équations de Hamilton, Bull. Acad. Roy. de Belg. 17 (1931).

The term “Lie derivative” (Liesche Ableitung) is due to van Dantzig,
who also suggested a definition using the flow of a vector field:

* D. van Dantzig, Zur allgemeinen projektiven Differentialgeometrie, Proc. Roy. Acad. Amsterdam 35 (1932) Part I: 524–534; Part II: 535–542.

An introduction in the context of [[synthetic differential geometry]] is in

* [[Gonzalo Reyes]], _Lie derivatives, Lie brackets and vector fields over curves_, [pdf](https://marieetgonzalo.files.wordpress.com/2006/10/lie-derivatives.pdf)

A gentle elementary introduction for mathematical physicists

* Bernard F. Schutz, *Geometrical methods of mathematical physics* (elementary intro) [amazon](http://www.amazon.com/Geometrical-Methods-Mathematical-Physics-Bernard/dp/0521298873), [google](http://books.google.hr/books?id=HAPMB2e643kC&lpg=PP1&dq=schutz%20geometrical%20methods&pg=PP1#v=onepage&q&f=false)

[[!redirects Lie derivatives]].


Lie differentiation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Integration theory
+--{: .hide}
[[!include integration theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Lie differentiation_  is the process reverse to [[Lie integration]]. It sends a [[Lie group]] to its [[Lie algebra]] and more generally a [[Lie groupoid]] to its [[Lie algebroid]] and a [[smooth ∞-group]] to its [[L-∞ algebra]].

* For the moment, for ordinary [[Lie theory]] see at _[[Lie's three theorems]]_.

* For [[infinity-Lie theory]] see at _[synthetic differential infinity-groupoid -- Lie differentiation](synthetic+differential+infinity-groupoid#LieDifferentiation)_ .

## Definition

A formalization of the notion Lie differentiation in [[higher geometry]] has been given in ([Lurie](#Lurie)), inspired by and building on results discussed at _[[model structure for L-∞ algebras]]_. This we discuss in

* [For deformation contexts](#ForDeformationContexts)

We then specialize this to those _[[deformation contexts]]_, def. \ref{DeformationContext}, that arise in the formalization of [[higher differential geometry]] by [[differential cohesion]]:

* [For cohesive contexts](#ForCohesiveContexts)

This is the context in which one has a natural formulation of ordinary Lie differentiation of ordinary [[Lie groups]] to [[Lie algebras]] and its generalization to the Lie differentiation of [[smooth ∞-groups]] to [[L-∞ algebras]]. See the discussion of _[Examples](#Examples)_ below for more.

### For deformation contexts
 {#ForDeformationContexts}

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

A **[[deformation context]]** is an [[(∞,1)-category]] $Sp_*$ such that

1. it is a [[presentable (∞,1)-category]];

1. it contains an [[initial object in an (∞,1)-category|initial object]]

and

* equipped with a [[set]] of [[objects]]

  $$
    \{E_\alpha \in Stab(Sp_*^{op})\}
  $$

  in the [[stabilization]] of the [[opposite (∞,1)-category]] $Sp_*^{op}$.

=--

This is ([Lurie, def. 1.1.3](#Lurie)) together with the assumption of a terminal object in $Sp_*^{op}$ stated on p.9 (and later implicialy used).

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

Definition \ref{DeformationContext} is meant to be read as follows:

We think of $Sp_*$ as an [[(∞,1)-category]] of [[pointed object|pointed]] [[spaces]] in some [[higher geometry]]. The point is the initial object.
We think of the formal duals of the objects $\{E_\alpha\}_\alpha$ as a set of generating [[infinitesimally thickened points]] (points in [[formal geometry]]).

=--

The following construction generates the "[[jets]]" induced by the generating infinitesimally thickened points.

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

Given a [[deformation context]] $(Sp_*, \{E_\alpha\}_\alpha)$, we say

* a [[morphism]] in $Sp_*^{op}$ is an **elementary morphism** if it is the [[homotopy fiber]] to a map into $\Omega^{\infty -n}E_\alpha$ for some $n \in \mathbb{Z}$ and some $\alpha$;

* a morphism in $Sp_*^{op}$ is a **small morphism** if it is the composite of [[finite set|finitely]] many elementary morphisms.

We write

$$
  Sp_*^{inf} \hookrightarrow Sp_*
$$

for the full [[sub-(∞,1)-category]] on those objects $A$ for which the essentially unique map $A \to *$ is small.

=--

([Lurie, def. 1.1.8](#Lurie))

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

Given a [[deformation context]] $(Sp_*, \{E_\alpha\}_\alpha)$, def. \ref{DeformationContext}, the [[(∞,1)-category]] of **[[formal moduli problems]]** over it is the full [[sub-(∞,1)-category]] of the [[(∞,1)-category of (∞,1)-presheaves]] over $Sp_*^{inf}$

$$
  FormalModuli^{Sp_*}
  \hookrightarrow
  PSh_\infty(Sp_*^{inf})
$$

on those [[(∞,1)-functors]] $X \colon (Sp_*^{inf})^{op} \to \infty Grpd$ such that

1. over the [[terminal object]] they are [[contractible]]: $X(*) \simeq *$;

1. they sends [[(∞,1)-colimits]] in $Sp_*^{inf}$ to [[(∞,1)-limits]] in [[∞Grpd]].

=--

([Lurie, def. 1.1.14](#Lurie))

([Lurie, remark 1.1.7](#Lurie)).

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

This means that a "formal deformation problem" is a [[space]] in [[higher geometry]] whose geometric structure is detected by the "test spaces" in $Sp_*^{inf}$ in a way that respects gluing ([[descent]]) in $Sp_*^{inf}$ as given by [[(∞,1)-colimits]] there. The first condition requires that there is an essentially unique such probe by the point, hence that these higher geometric space have essentially a single global point. This is the condition that reflects the infinitesimal nature of the deformation problem.

=--

We will often just write $Sp_*$ for a deformation context $(Sp_*, \{E_\alpha\}_\alpha)$, when the objects $\{E_\alpha\}$ are understood.

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

The [[(∞,1)-category]] $FormalModuli^{Sp_*}$ of [[formal moduli problems]] is a [[presentable (∞,1)-category]]. Moreover it is a [[reflective sub-(∞,1)-category]] of the [[(∞,1)-category of (∞,1)-presheaves]]

$$
  FormalModuli^{Sp_*}
  \stackrel{\leftarrow}{\hookrightarrow}
  PSh_{\infty}(Sp_*)
  \,.
$$

=--

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

Given a [[deformation context]] $Sp_*$, the restricted [[(∞,1)-Yoneda embedding]] gives an [[(∞,1)-functor]]

$$
  Lie \;\colon\; Sp_* \to FormalModuli^{Sp_*}
  \,.
$$

=--

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

For $(X,x) \in Sp_*$, the object $Lie(X,x)$ represents the _[[formal geometry|formal neighbourhood]]_ of the basepoint $x$ of $X$ as seen by the infinitesimally thickened points dual to the $\{E_\alpha\}$.

Hence we may call this the operation of **Lie differentiation** of spaces in $Sp_*$ around their given base point.

In the archetypical implementation of these axiomatics, discuss [below](#Examples), there is an [[equivalence of (∞,1)-categories]] of [[formal moduli problems]] with [[L-∞ algebras]] and the Lie differentiation of the [[delooping]]/[[moduli ∞-stack]] $\mathbf{B}G$ of a [[smooth ∞-group]] $G$ is its [[L-∞ algebra]] $\mathfrak{g}$: $Lie(\mathbf{B}G) \simeq \mathbf{B}\mathfrak{g}$.

=--

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

The Lie differentiation functor

$$
  Lie
  \;
  \colon
  \;
  Sp_* \to FormalModuli^{Sp_*}
$$

of prop. \ref{LieDifferentiationFunctor} preserves [[(∞,1)-limits]].

=--

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

By prop. \ref{FormalModuliIsPresentableCategory} the [[(∞,1)-limits]] in $FormalModuli^{Sp_*}$ may be computed in $PSh_\infty(Sp_*)$.  There the statement is that of the [[(∞,1)-Yoneda embedding]], or rather just the statement that the [[derived hom space|(∞,1)-hom (∞,1)-functor]] $Sp_*(D,-)$ preserves $(\infty,1)$-limits.

=--

### For cohesive contexts
 {#ForCohesiveContexts}

> under construction

Let $\mathbf{H}_{th}$ be a [[cohesive (∞,1)-topos]] $(ʃ \dashv \flat \dashv \sharp)$ equipped with
[[differential cohesion]] $(Red \dashv ʃ_{inf} \dashv \flat_{inf})$.

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

A set of objects $\{D_\alpha \in \mathbf{H}_{th}\}_\alpha$
is said to **exhibit the differential structure** or
**exhibit the infinitesimal thickening** of $\mathbf{H}_{th}$ if the
[[localization of an (∞,1)-category|localization]]

$$
  L_{\{D_\alpha\}} \mathbf{H}_{th}
  \stackrel{\leftarrow}{\hookrightarrow}
  \mathbf{H}_{th}
$$

of $\mathbf{H}_{th}$
at the morphisms of the form $D_\alpha \times X \to X$
is exhibited by the [[infinitesimal shape modality]] $ʃ_{inf} \coloneqq i^* i_*$

$$
  \mathbf{H}
  \simeq
  L_{\{D_\alpha\}} \mathbf{H}_{th}
  \stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\hookrightarrow}}
  \mathbf{H}_{th}
  \,.
$$

=--

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

Def. \ref{ExhibitingInfinitesimalCohesion} expresses
the [[infinitesimal cohesion|infinitesimal]]
analog of the notion of objects exhibiting [[cohesion]],
see at structures in cohesion -- A1-homotopy and the continuum,
hence an infinitesimal notion of [[A1-homotopy theory]].

=--

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

If objects $\{D_\alpha \in \mathbf{H}_{th}\}$ exhibit the differential cohesion of $\mathbf{H}_{th}$,
then they are essentially uniquely pointed.

=--

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

The localizing objects are in particular themselves [[local objects]]
so that $ʃ_{inf} D_\alpha \simeq *$.
By the $(Red \dashv ʃ_{inf})$-[[adjunctions]] this means that

$$
  \begin{aligned}
    \mathbf{H}_{th}(*, D_\alpha)
    & \simeq \mathbf{H}_{th}(Red(*), D_\alpha)
    \\
    & \simeq \mathbf{H}_{th}(*, ʃ_{inf} D_\alpha)
    \\
    & \simeq \mathbf{H}_{th}(*, *)
    \\
    & \simeq *
  \end{aligned}
  \,.
$$

=--

We now consider $(\mathbf{H}_{th}^{\ast/}, \{D_\alpha\})$
as a [[deformation context]], def. \ref{DeformationContext}.

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

Write

$$
  Lie
  \;\colon\;
  \mathbf{H}_{th}^{\ast/}
  \to
  FormalModuli^{\mathbf{H}_{th}^{*/}}
  \hookrightarrow
  PSh_\infty(\mathbf{H}_{th}^{*/})
$$

for the Lie differentiaon [[(∞,1)-functor]], def. \ref{LieDifferentiationFunctor},
which sends $(x \colon * \to X) \in \mathbf{H}_{th}$ to

$$
  Lie(X,x)
   \;\colon\;
  D
  \mapsto \mathbf{H}^{\ast/}(D,(X,x))
  \,.
$$

=--

## Examples
 {#Examples}

### Examples of contexts for Lie differentiation

#### dg-Geometry

[[dg-geometry]] (the running example in ([Lurie](#Lurie))).

#### Synthetic-differential $\infty$-groupoids
 {#ExampleContextSyntheticDifferential}

* [[SynthDiff∞Grpd]]

(...)

_[synthetic differential infinity-groupoid -- Lie differentiation](synthetic+differential+infinity-groupoid#LieDifferentiation)_

(...)

### Examples of Lie differentiation

#### Of a Lie group

(...)

#### Of a Lie groupoid

Given a Lie groupoid $G_1\Rightarrow G_0$, we take the vector bundle $ker Ts|_{G_0}$ restricted on $G_0$, then we show that there is a [[Lie algebroid]] structure on $A:=ker Ts|_{G_0} \to G_0$. First of all, the anchor map is given by $ker Ts_{G_0} \xrightarrow{Tt} G_0$. Secondly, to define the Lie bracket, one shows that a section $X$ of $A$ may be right translated to a vector field on $G_1$, which is right invariant. Then Jacobi identity implies that right invariant vector fields are closed under Lie bracket. Thus Lie brackets on vector fields on $G_1$ induces a Lie bracket on sections of $A$.

## Related concepts

[[!include infinitesimal and local - table]]

## References

Lie differentiation of [[Lie n-groupoids]] was first considered in generality in

* [[Pavol Ševera]], _$L_\infty$ algebras as 1-jets of simplicial manifolds (and a bit beyond)_ ([arXiv:0612349](http://arxiv.org/abs/math/0612349))

See also theorem 8.28 of

* Du Li, _Higher Groupoid Actions, Bibundles, and Differentiation_ ([arXiv:1512.04209](http://arxiv.org/abs/1512.04209))

Lie differentiation [in deformation contexts](#ForDeformationContexts) is formulated in section 1 of

* {#Lurie} [[Jacob Lurie]], _[[Formal moduli problems]]_