Lie algebra cohomology

(Redirected from Cohomology of Lie algebras)

In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was defined by Claude Chevalley and Samuel Eilenberg (1948) in order to give an algebraic construction of the cohomology of the underlying topological spaces of compact Lie groups. In the paper above, a specific chain complex, called the Koszul complex, is defined for a module over a Lie algebra, and its cohomology is taken in the normal sense.

Motivation

If ${\displaystyle G}$ 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 ${\displaystyle G}$. 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 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.

It should be noted that if ${\displaystyle G}$ is a simply connected noncompact Lie group, the Lie algebra cohomology of the associated Lie algebra ${\displaystyle {\mathfrak {g}}}$ does not necessarily reproduce the de Rham cohomology of ${\displaystyle G}$. 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.

Definition

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

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

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

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

Analogously, one can define Lie algebra homology as

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

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

${\displaystyle M\mapsto M_{\mathfrak {g}}:=M/{\mathfrak {g}}M.}$

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

Chevalley-Eilenberg complex

The Lie algebra cohomology of the Lie algebra ${\displaystyle {\mathfrak {g}}}$ over the field ${\displaystyle k}$, with values in the left ${\displaystyle {\mathfrak {g}}}$-module ${\displaystyle M}$ can be computed using the Chevalley-Eilenberg complex ${\displaystyle \mathrm {Hom} _{k}(\Lambda ^{\ast }{\mathfrak {g}},M)}$. The ${\displaystyle n}$-cochains in this complex are the alternating ${\displaystyle k}$-multilinear functions ${\displaystyle f:\Lambda ^{n}{\mathfrak {g}}\to M}$ of ${\displaystyle n}$ variables with values in ${\displaystyle M}$. The coboundary of an ${\displaystyle n}$-cochain is the ${\displaystyle (n+1)}$-cochain ${\displaystyle \delta f}$ given by[1]

{\displaystyle {\begin{aligned}(\delta 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

where the caret signifies omitting that argument.

Cohomology in small dimensions

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

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

The first cohomology group is the space Der of derivations modulo the space Ider of inner derivations

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

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

${\displaystyle d[x,y]=xdy-ydx~}$

and is called inner if it is given by

${\displaystyle dx=xa~}$

for some a in M.

The second cohomology group

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

is the space of equivalence classes of Lie algebra extensions

${\displaystyle 0\rightarrow M\rightarrow {\mathfrak {h}}\rightarrow {\mathfrak {g}}\rightarrow 0}$

of the Lie algebra by the module M.

There do not seem to be any similar easy interpretations for the higher cohomology groups.