Special and general types
∞-Lie theory (higher geometry)
Formal Lie groupoids
Lie algebra cohomology is the intrinsic notion of cohomology of Lie algebras.
There is a precise sense in which Lie algebras 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.
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 -Lie algebra with coefficients in the left -module is defined as where is the ground field understood as a trivial module over the universal enveloping algebra . In particular it is a derived functor.
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 is naturally equipped with a differential to start with (see below).
The first argument in the Hom, i.e. is sometimes called the Chevalley-Eilenberg chain complex (cf. Weibel); the Chevalley-Eilenberg cochain complex is the whole thing, i.e.
If is a trivial module then and if is finite-dimensional this equals with an appropriate differential and the exterior multiplication gives it a dg-algebra structure.
Via -Lie algebras
As discussed at Chevalley-Eilenberg algebra, we may identify Lie algebras as the duals of dg-algebras whose underlying graded algebra is the Grassmann algebra on the vector space .
Similarly, a dg-algebra whose underlying algebra is free on a graded vector space we may understand as exibiting an ∞-Lie algebra-structure on .
Then a morphism of these -Lie algebras is by definition just a morphism of dg-algebras. Such a morphis may be thought of as a cocycle in nonabelian Lie algebra cohomology .
Specifically, write for the line Lie n-algebra, the -Lie algebra given by the fact that has a single generator in degree and vanishing differential. Then a morphism
is a cocycle in the abelian Lie algebra cohomology . Notice that dually, by definition, this is a morphism of dg-algebras
Since on the right we only have a single closed degree- generator, such a morphism is precily a closed degree -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.
The following lemma asserts that for semisimple Lie algebras only the cohomology with coefficients in the trivial module is nontrivial.
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.
Every invariant polynomial on a Lie algebra has a transgression to a cocycle on . See ∞-Lie algebra cohomology for more.
For instance for a semisimple Lie algebra, there is the Killing form . The corresponding 3-cocycle is
that is: the function that sends three Lie algebra elements to the number .
On the super Poincare Lie algebra in dimension (10,1) there is a 4-cocycle
Every Lie algebra degree cocycle (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 -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.
Ordinary Lie algebras
An account of the standard theory of Lie algebra cohomology is for instance
in chapter V in vol III of
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
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 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) .
The ∞-Lie algebra extensions induced by a degree -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