Lie algebra cohomology#

If $G$ is a compact simply connected Lie group, then it is determined by its Lie algebra, so it should be possible to calculate its cohomology from the Lie algebra. This can be done as follows. Its cohomology is the de Rham cohomology of the complex of differential forms on $G$. Using an averaging process, this complex can be replaced by the complex of left-invariant differential forms. The left-invariant forms, meanwhile, are determined by their values at the identity, so that the space of left-invariant differential forms can be identified with the exterior algebra of the Lie algebra, with a suitable differential.

The construction of this differential on an exterior algebra makes sense for any Lie algebra, so it is used to define Lie algebra cohomology for all Lie algebras. More generally one uses a similar construction to define Lie algebra cohomology with coefficients in a module.

If $G$ is a simply connected noncompact Lie group, the Lie algebra cohomology of the associated Lie algebra $\backslash mathfrak\; g$ does not necessarily reproduce the de Rham cohomology of $G$. The reason for this is that the passage from the complex of all differential forms to the complex of left-invariant differential forms uses an averaging process that only makes sense for compact groups.

Let $\backslash mathfrak\; g$ be a Lie algebra over a commutative ring R with universal enveloping algebra $U\backslash mathfrak\; g$, and let M be a representation of $\backslash mathfrak\; g$ (equivalently, a $U\backslash mathfrak\; g$-module). Considering R as a trivial representation of $\backslash mathfrak\; g$, one defines the cohomology groups

:$\backslash mathrm\{H\}^n(\backslash mathfrak\{g\};\; M)\; :=\; \backslash mathrm\{Ext\}^n\_\{U\backslash mathfrak\{g\}\}(R,\; M)$

(see Ext functor for the definition of Ext). Equivalently, these are the right derived functors of the left exact invariant submodule functor

:$M\; \backslash mapsto\; M^\{\backslash mathfrak\{g\}\}\; :=\; \backslash \{\; m\; \backslash in\; M\; \backslash mid\; xm\; =\; 0\backslash \; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; \backslash mathfrak\{g\}\backslash \}.$

Analogously, one can define Lie algebra homology as

:$\backslash mathrm\{H\}\_n(\backslash mathfrak\{g\};\; M)\; :=\; \backslash mathrm\{Tor\}\_n^\{U\backslash mathfrak\{g\}\}(R,\; M)$

(see Tor functor for the definition of Tor), which is equivalent to the left derived functors of the right exact coinvariants functor

:$M\; \backslash mapsto\; M\_\{\backslash mathfrak\{g\}\}\; :=\; M\; /\; \backslash mathfrak\{g\}\; M.$

Some important basic results about the cohomology of Lie algebras include Whitehead's lemmas, Weyl's theorem, and the Levi decomposition theorem.

Let $\backslash mathfrak\{g\}$ be a Lie algebra over a field $k$, with a left action on the $\backslash mathfrak\{g\}$-module $M$. The elements of the Chevalley–Eilenberg complex

: $\backslash mathrm\{Hom\}\_k(\backslash Lambda^\backslash bullet\backslash mathfrak\{g\},M)$

are called cochains from $\backslash mathfrak\{g\}$ to $M$. A homogeneous $n$-cochain from $\backslash mathfrak\{g\}$ to $M$ is thus an alternating $k$-multilinear function $f\backslash colon\backslash Lambda^n\backslash mathfrak\{g\}\backslash to\; M$. When $\backslash mathfrak\{g\}$ is finitely generated as vector space, the Chevalley–Eilenberg complex is canonically isomorphic to the tensor product $M\; \backslash otimes\; \backslash Lambda^\{\backslash bullet\}\backslash mathfrak\{g\}^*$, where $\backslash mathfrak\{g\}^*$denotes the dual vector space of $\backslash mathfrak\{g\}$.

The Lie bracket $[\backslash cdot,\backslash cdot]\backslash colon\; \backslash Lambda^2\; \backslash mathfrak\{g\}\; \backslash rightarrow\; \backslash mathfrak\{g\}$ on $\backslash mathfrak\{g\}$ induces a transpose application $d^\{(1)\}\_\{\backslash mathfrak\{g\}\}\; \backslash colon\; \backslash mathfrak\{g\}^*\; \backslash rightarrow\; \backslash Lambda^2\; \backslash mathfrak\{g\}^*$ by duality. The latter is sufficient to define a derivation $d\_\{\backslash mathfrak\{g\}\}$ of the complex of cochains from $\backslash mathfrak\{g\}$ to $k$ by extending $d\_\{\backslash mathfrak\{g\}\}^\{(1)\}$according to the graded Leibniz rule. It follows from the Jacobi identity that $d\_\{\backslash mathfrak\{g\}\}$ satisfies $d\_\{\backslash mathfrak\{g\}\}^2\; =\; 0$ and is in fact a differential. In this setting, $k$ is viewed as a trivial $\backslash mathfrak\{g\}$-module while $k\; \backslash sim\; \backslash Lambda^0\backslash mathfrak\{g\}^*\; \backslash subseteq\; \backslash mathrm\{Ker\}(d\_\{\backslash mathfrak\{g\}\})$ may be thought of as constants.

In general, let $\backslash gamma\; \backslash in\; \backslash mathrm\{Hom\}(\backslash mathfrak\{g\},\; \backslash operatorname\{End\}\; M)$ denote the left action of $\backslash mathfrak\{g\}$ on $M$ and regard it as an application $d\_\backslash gamma^\{(0)\}\; \backslash colon\; M\; \backslash rightarrow\; M\; \backslash otimes\; \backslash mathfrak\{g\}^*$. The Chevalley–Eilenberg differential $d$ is then the unique derivation extending $d\_\backslash gamma^\{(0)\}$ and $d\_\{\backslash mathfrak\{g\}\}^\{(1)\}$ according to the graded Leibniz rule, the nilpotency condition $d^2\; =\; 0$ following from the Lie algebra homomorphism from $\backslash mathfrak\{g\}$ to $\backslash operatorname\{End\}\; M$ and the Jacobi identity in $\backslash mathfrak\{g\}$.

Explicitly, the differential of the $n$-cochain $f$ is the $(n+1)$-cochain $df$ given by:{{cite book|title=An introduction to homological algebra|last1=Weibel|first1=Charles A.|author-link=Charles Weibel|date=1994|publisher=Cambridge University Press|page=240}}

:$\backslash begin\{align\}$

(d f)\left(x_1, \ldots, x_{n+1}\right) =

&\sum_i (-1)^{i+1}x_i\, f\left(x_1, \ldots, \hat x_i, \ldots, x_{n+1}\right) + \\

&\sum_{i

\end{align}

where the caret signifies omitting that argument.

When $G$ is a real Lie group with Lie algebra $\backslash mathfrak\{g\}$, the Chevalley–Eilenberg complex may also be canonically identified with the space of left-invariant forms with values in $M$, denoted by $\backslash Omega^\{\backslash bullet\}(G,M)^G$. The Chevalley–Eilenberg differential may then be thought of as a restriction of the covariant derivative on the trivial fiber bundle $G\; \backslash times\; M\; \backslash rightarrow\; G$, equipped with the equivariant connection $\backslash tilde\{\backslash gamma\}\; \backslash in\; \backslash Omega^1(G,\; \backslash operatorname\{End\}\; M)$ associated with the left action $\backslash gamma\; \backslash in\; \backslash mathrm\{Hom\}(\backslash mathfrak\{g\},\; \backslash operatorname\{End\}\; M)$ of $\backslash mathfrak\{g\}$ on $M$. In the particular case where $M\; =\; k\; =\; \backslash mathbb\{R\}$ is equipped with the trivial action of $\backslash mathfrak\{g\}$, the Chevalley–Eilenberg differential coincides with the restriction of the de Rham differential on $\backslash Omega^\{\backslash bullet\}(G)$ to the subspace of left-invariant differential forms.

The zeroth cohomology group is (by definition) the invariants of the Lie algebra acting on the module:

:$H^0(\backslash mathfrak\{g\};\; M)\; =\; M^\{\backslash mathfrak\{g\}\}\; =\; \backslash \{\; m\; \backslash in\; M\; \backslash mid\; xm\; =\; 0\backslash \; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; \backslash mathfrak\{g\}\backslash \}.$

The first cohomology group is the space {{math|Der}} of derivations modulo the space {{math|Ider}} of inner derivations

:$H^1(\backslash mathfrak\{g\};\; M)\; =\; \backslash mathrm\{Der\}(\backslash mathfrak\{g\},\; M)/\backslash mathrm\{Ider\}\; (\backslash mathfrak\{g\},\; M)\backslash ,$,

where a derivation is a map $d$ from the Lie algebra to $M$ such that

:$d[x,y]\; =\; xdy-ydx~$

and is called inner if it is given by

:$dx\; =\; xa~$

for some $a$ in $M$.

The second cohomology group

:$H^2(\backslash mathfrak\{g\};\; M)$

is the space of equivalence classes of Lie algebra extensions

:$0\backslash rightarrow\; M\backslash rightarrow\; \backslash mathfrak\{h\}\backslash rightarrow\backslash mathfrak\{g\}\backslash rightarrow\; 0$

of the Lie algebra by the module $M$.

Similarly, any element of the cohomology group $H^\{n+1\}(\backslash mathfrak\{g\};\; M)$ gives an equivalence class of ways to extend the Lie algebra $\backslash mathfrak\{g\}$ to a "Lie $n$-algebra" with $\backslash mathfrak\{g\}$ in grade zero and $M$ in grade $n$.{{Cite journal|last1=Baez|first1=John C.|last2=Crans|first2=Alissa S.|author-link1=John C. Baez|date=2004|title=Higher-dimensional algebra VI: Lie 2-algebras |journal=Theory and Applications of Categories|volume=12|pages=492–528 |arxiv=math/0307263 |citeseerx=10.1.1.435.9259 |bibcode=2003math......7263B }} A Lie $n$-algebra is a homotopy Lie algebra with nonzero terms only in degrees 0 through $n$.

When $M\; =\; \backslash mathbb\{R\}$, as mentioned earlier the Chevalley–Eilenberg complex coincides with the de-Rham complex for a corresponding compact Lie group. In this case $M$ carries the trivial action of $\backslash mathfrak\{g\}$, so $xa\; =\; 0$ for every $x\; \backslash in\; \backslash mathfrak\{g\},\; a\; \backslash in\; M$.

• The zeroth cohomology group is $M$.
• First cohomology: given a derivation $D$, $xDy\; =\; 0$ for all $x$ and $y$, so derivations satisfy $D([x,y])\; =\; 0$ for all commutators, so the ideal $[\backslash mathfrak\{g\},\; \backslash mathfrak\{g\}]$ is contained in the kernel of $D$.
• If $[\backslash mathfrak\{g\},\; \backslash mathfrak\{g\}]\; =\; \backslash mathfrak\{g\}$, as is the case for simple Lie algebras, then $D\; \backslash equiv\; 0$, so the space of derivations is trivial, so the first cohomology is trivial.
• If $\backslash mathfrak\{g\}$ is abelian, that is, $[\backslash mathfrak\{g\},\; \backslash mathfrak\{g\}]\; =\; 0$, then any linear functional $D:\; \backslash mathfrak\{g\}\; \backslash rightarrow\; M$ is in fact a derivation, and the set of inner derivations is trivial as they satisfy $Dx\; =\; xa\; =\; 0$ for any $a\; \backslash in\; M$. Then the first cohomology group in this case is $M^\{\backslash text\{dim\}\backslash mathfrak\{g\}\}$. In light of the de-Rham correspondence, this shows the importance of the compact assumption, as this is the first cohomology group of the $n$-torus viewed as an abelian group, and $\backslash mathbb\{R\}^n$ can also be viewed as an abelian group of dimension $n$, but $\backslash mathbb\{R\}^n$ has trivial cohomology.
• Second cohomology: The second cohomology group is the space of equivalence classes of central extensions

:$$0\; \backslash rightarrow\; \backslash mathfrak\{h\}\; \backslash rightarrow\; \backslash mathfrak\{e\}\; \backslash rightarrow\; \backslash mathfrak\{g\}\; \backslash rightarrow\; 0.$$

Finite dimensional, simple Lie algebras only have trivial central extensions: a proof is provided here.

When $M\; =\; \backslash mathfrak\{g\}$, the action is the adjoint action, $x\backslash cdot\; y\; =\; [x,y]\; =\; \backslash text\{ad\}(x)y$.

• The zeroth cohomology group is the center $\backslash mathfrak\{z\}(\backslash mathfrak\{g\})$
• First cohomology: the inner derivations are given by $Dx\; =\; xy\; =\; [x,y]\; =\; -\backslash text\{ad\}(y)x$, so they are precisely the image of $\backslash text\{ad\}:\; \backslash mathfrak\{g\}\; \backslash rightarrow\; \backslash operatorname\{End\}\; \backslash mathfrak\{g\}.$ The first cohomology group is the space of outer derivations.

• BRST formalism in theoretical physics.
• Gelfand–Fuks cohomology

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{{refbegin}}

• {{Citation | last1=Chevalley | first1=Claude | author1-link=Claude Chevalley | last2=Eilenberg | first2=Samuel | author2-link=Samuel Eilenberg | title=Cohomology theory of Lie groups and Lie algebras | jstor=1990637 | publisher=American Mathematical Society | location=Providence, R.I. | mr=0024908 | year=1948 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=63 | issue=1 | pages=85–124 | doi=10.2307/1990637| doi-access=free }}
• {{Citation | last1=Hilton | first1=Peter J. | author-link1=Peter Hilton | last2=Stammbach | first2=Urs | title=A course in homological algebra | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-94823-2 | mr=1438546 | year=1997 | volume=4}}
• {{Citation | last1=Knapp | first1=Anthony W. | author-link1=Anthony W. Knapp | title=Lie groups, Lie algebras, and cohomology | publisher=Princeton University Press | series=Mathematical Notes | isbn=978-0-691-08498-5 | mr=938524 | year=1988 | volume=34}}

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Category:Cohomology theories

Category:Homological algebra

Category:Lie algebras