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 $H_{\mathrm{Lie}}^{*}(𝔤,M)={\mathrm{Ext}}_{U𝔤}^{*}(k,M)$ 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

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.

If $M$ is a trivial module $k$ then $\mathrm{CE}(𝔤):={\mathrm{Hom}}_k({\Lambda }^{*}𝔤,k)$ 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 $\mathrm{\infty }$-Lie algebras

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

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

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

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

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

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

Examples

Every invariant polynomial $⟨-⟩\in W(𝔤)$ 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

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

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

Extensions

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

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

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

• 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

• infinity-Lie algebra cohomology

References

Ordinary Lie algebras

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

in chapter V in vol III of

• Werner Greub, Stephen Halperin, Ray Vanstone, Connections, Curvature, and Cohomology Academic Press (1973)

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

• eom Lie algebra cohomology

• wikipedia

• scholarpedia: An introduction to Lie algebra cohomology

Super Lie algebras

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

• John Baez, John Huerta, Division algebras and supersymmetry I (arXiv:0909.0551)

• John Baez, John Huerta, Division algebras and supersymmetry II (arXiv:1003.34360)

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