nLab Lie algebra cohomology

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.

Hochschild-Serre spectral sequence

Definition

(relative Lie algebra cohomology with coefficents)

Let

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

2. $(V, \rho)$ a $\mathfrak{g}$-Lie algebra module of finite dimension;

3. $\mathfrak{h} \hookrightarrow \mathfrak{g}$ 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 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)

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

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

Definition

(Lie algebra reductive in ambient Lie algebra)

A sub-Lie algebra

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

is called reductive if the adjoint Lie algebra representation of $\mathfrak{h}$ on $\mathfrak{g}$ is reducible.

(Koszul 50, recalled in e.g. Solleveld 02, def. 2.27)

Proposition

(invariants in Lie algebra cohomology computed by relative Lie algebra cohomology)

Let

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

2. $(V, \rho)$ a $\mathfrak{g}$-Lie algebra module of finite dimension, which is reducible;

3. $\mathfrak{h} \hookrightarrow \mathfrak{g}$ a sub-Lie algebra which is reductive in $\mathfrak{g}$ (Def. ) in that its adjoint representation on $\mathfrak{g}$ is reducible.

4. 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. , using the invariant subcomplex!):

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

(Solleveld 02, theorem 2.28)

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 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

• 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

• 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)

• 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]

• 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, pdf]

• 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, eudml:40921]

• Maarten Solleveld, Lie algebra cohomology and Macdonald’s conjectures, 2002 (pdf)

See also

• eomLie 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

• 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)

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

  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) .

On Lie algebra cohomology of super Lie algebras (see also the brane scan) in relation to 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)

• C. A. Cremonini, Pietro Antonio Grassi, Cohomology of Lie Superalgebras: Forms, Pseudoforms, and Integral Forms (arXiv:2106.11786)

Extensions

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

• 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