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 $\mathfrak{g}$ 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.

## 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 algebras $\mathfrak{g}$ as the duals $CE(\mathfrak{g})$ of dg-algebras 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.

###### 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.

## 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

$\mu = \langle -, [-,-] \rangle : CE(\mathfrak{g}) \,,$

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

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 algebras this was observed in (BaezCrans Theorem 55).

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 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

• José 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

• José de Azcárraga, José 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 $\leq 5$ is analyzed in

• 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 $b^{n-2} \to \mathfrak{g}_\mu \to \mathfrak{g}$ 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 April 28, 2015 12:07:39 by Urs Schreiber (195.113.30.252)