cohomology

∞-Lie theory

# Contents

## Idea

Lie algebra cohomology is the intrinsic notion of cohomology of Lie algebras.

There is a precise sense in which Lie algebras $𝔤$ are infinitesimal Lie groups. 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.

## Defintion

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 $𝔤$ with coefficients in the left $𝔤$-module $M$ is defined as ${H}_{\mathrm{Lie}}^{*}\left(𝔤,M\right)={\mathrm{Ext}}_{U𝔤}^{*}\left(k,M\right)$ where $k$ is the ground field understood as a trivial module over the universal enveloping algebra $U𝔤$. 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

${\mathrm{Hom}}_{𝔤}\left(U𝔤{\otimes }_{k}{\Lambda }^{*}𝔤,M\right)\cong {\mathrm{Hom}}_{k}\left({\Lambda }^{*}𝔤,M\right),$Hom_{\mathfrak{g}}(U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g},M)\cong Hom_k(\Lambda^* \mathfrak{g},M),

where the first argument $U𝔤{\otimes }_{k}{\Lambda }^{*}𝔤$ is naturally equipped with a differential to start with (see below).

WHERE BELOW?

The first argument in the Hom, i.e. $U𝔤{\otimes }_{k}{\Lambda }^{*}𝔤$ is sometimes called the Chevalley-Eilenberg chain complex (cf. Weibel); the Chevalley-Eilenberg cochain complex is the whole thing, i.e.

$\mathrm{CE}\left(𝔤,M\right):={\mathrm{Hom}}_{𝔤}\left(U𝔤{\otimes }_{k}{\Lambda }^{*}𝔤,M\right)\cong {\mathrm{Hom}}_{k}\left({\Lambda }^{*}𝔤,M\right).$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 $\mathrm{CE}\left(𝔤\right):={\mathrm{Hom}}_{k}\left({\Lambda }^{*}𝔤,k\right)$ and if $𝔤$ is finite-dimensional this equals ${\Lambda }^{*}{𝔤}^{*}$ 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 algebras $𝔤$ as the duals $\mathrm{CE}\left(𝔤\right)$ of dg-algebras whose underlying graded algebra is the Grassmann algebra on the vector space ${𝔤}^{*}$.

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

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

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

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

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

$\mathrm{CE}\left(𝔤\right)←\mathrm{CE}\left({b}^{n-1}ℝ\right):\mu \phantom{\rule{thinmathspace}{0ex}}.$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

$\mu \in \mathrm{CE}\left(𝔤\right)\phantom{\rule{thinmathspace}{0ex}}.$\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

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

Whiteheads lemma

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

${H}^{p}\left(𝔤,V\right)=0\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{for}\phantom{\rule{thickmathspace}{0ex}}p>0\phantom{\rule{thinmathspace}{0ex}}.$H^p(\mathfrak{g}, V) = 0 \;\;\; for\;p \gt 0 \,.

## Examples

Every invariant polynomial $⟨-⟩\in W\left(𝔤\right)$ on a Lie algebra has a transgression to a cocycle on $𝔤$. See ∞-Lie algebra cohomology for more.

For instance for $𝔤$ a semisimple Lie algebra, there is the Killing form $⟨-,-⟩$. The corresponding 3-cocycle is

$\mu =⟨-,\left[-,-\right]⟩:\mathrm{CE}\left(𝔤\right)\phantom{\rule{thinmathspace}{0ex}},$\mu = \langle -, [-,-] \rangle : CE(\mathfrak{g}) \,,

that is: the function that sends three Lie algebra elements $x,y,z$ to the number $\mu \left(x,y,z\right)=⟨x,\left[y,z\right]⟩$.

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

${\mu }_{4}=\overline{\psi }\wedge {\Gamma }^{ab}\Psi \wedge {e}_{a}\wedge {e}_{b}\in \mathrm{CE}\left(\mathrm{𝔰𝔦𝔰𝔬}\left(10,1\right)\right)$\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

${b}^{n-2}ℝ\to {𝔤}_{\mu }\to 𝔤\phantom{\rule{thinmathspace}{0ex}}.$b^{n-2} \mathbb{R} \to \mathfrak{g}_{\mu} \to \mathfrak{g} \,.

In the language of ∞-Lie algebras this was observed in (BaezCrans Theorem 55).

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

$\mathrm{CE}\left({𝔤}_{\mu }\right)←\mathrm{CE}\left(𝔤\right)$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 algebras many useful statements in ∞-Lie theory are obtained.

Examples

## References

### Ordinary Lie algebras

An account of the standard theory of Lie algebra cohomology is for instance

in chapter V in vol III of

in section 6 of

• J. A. de Azcárraga, José M. Izquierdo, Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics , Cambridge monographs of mathematical physics, (1995)

with a brief summary in

• J. A. de Azcarraga, J. M. Izquierdo, J. C. Perez Bueno, An introduction to some novel applications of Lie algebra cohomology and physics (arXiv)

chapter 7 of

• Charles Weibel, An introduction to homological algebra, Cambridge Studies in Adv. Math. 38, CUP 1994

See also

### Super Lie algebras

The cohomology of super Lie algebras is analyzed via normed division algebras in

See also division algebra and supersymmetry.

This subsumes some of the results in

• J. A. de Azcárraga and P. K. Townsend, Superspace geometry and classification of supersymmetric extended objects, Phys. Rev. Lett. 62, 2579–2582 (1989)

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

• Friedemann Brandt?,

Supersymmetry algebra cohomology I: Definition and general structure J. Math. Phys.51:122302, 2010, arXiv

Supersymmetry algebra cohomology II: Primitive elements in 2 and 3 dimensions J. Math. Phys. 51 (2010) 112303 (arXiv)

Supersymmetry algebra cohomology III: Primitive elements in four and five dimensions (arXiv)

and in higher dimensions more generally in

• Michael Movshev, Albert Schwarz, Renjun Xu, Homology of Lie algebra of supersymmetries (arXiv) .

### Extensions

The ∞-Lie algebra extensions ${b}^{n-2}\to {𝔤}_{\mu }\to 𝔤$ induced by a degree $n$-cocycle are considered around theorem 55 in

• John Baez and Alissa Crans, Higher-Dimensional Algebra VI: Lie 2-Algebras, Theory and Applications of Categories 12 (2004), 492-528. arXiv

Revised on August 2, 2012 16:40:08 by jim stasheff (98.114.103.37)