∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
$\infty$-Lie groupoids
$\infty$-Lie groups
$\infty$-Lie algebroids
$\infty$-Lie algebras
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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.
There are several different but equivalent definitions of the cohomology of a Lie algebra.
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.
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 $U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g}$ 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.
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.
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
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
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.
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.
(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
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.
(relative Lie algebra cohomology with coefficents)
Let
$(\mathfrak{g}, [-,-])$ be a Lie algebra of finite dimension;
$(V, \rho)$ a $\mathfrak{g}$-Lie algebra module of finite dimension;
$\mathfrak{h} \hookrightarrow \mathfrak{g}$ a sub-Lie algebra.
Consider the $\mathbb{N}$-graded vector space
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
The resulting cochain complex is the Chevalley-Eilenberg complex of $\mathfrak{g}$ relative $\mathfrak{h}$ with coefficients in $V$.
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:
(Lie algebra reductive in ambient Lie algebra)
A sub-Lie algebra
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)
(invariants in Lie algebra cohomology computed by relative Lie algebra cohomology)
Let
$(\mathfrak{g}, [-,-])$ be a Lie algebra of finite dimension;
$(V, \rho)$ a $\mathfrak{g}$-Lie algebra module of finite dimension, which is reducible;
$\mathfrak{h} \hookrightarrow \mathfrak{g}$ a sub-Lie algebra which is reductive in $\mathfrak{g}$ (Def. ) in that its adjoint representation on $\mathfrak{g}$ is reducible.
such that
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!):
Proof via the Hochschild-Serre spectral sequence.
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
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
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.
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, (1995)
José de Azcárraga, José M. Izquierdo, J. C. Perez Bueno, An introduction to some novel applications of Lie algebra cohomology and physics (arXiv:physics/9803046)
Maarten Solleveld, Lie algebra cohomology and
Macdonald’s conjectures_, 2002 (pdf)
See also
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
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
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)
The ∞-Lie algebra extensions $b^{n-2} \to \mathfrak{g}_\mu \to \mathfrak{g}$ induced by a degree $n$-cocycle are considered
Last revised on June 23, 2021 at 06:15:37. See the history of this page for a list of all contributions to it.