In mathematics, Lie algebra cohomology is a cohomology

Cohomology

In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...

 theory for Lie algebras. It was defined by in order to give an algebraic construction of the cohomology of the underlying topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

s of compact Lie groups. In the paper above, a specific complex, called the Koszul complex

Koszul complex

In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul...

, is defined for a module

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

 over a Lie algebra, and its cohomology is taken in the normal sense.

Motivation

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

De Rham cohomology

In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes...

 of the complex of differential forms on G. This can be replaced by the complex of equivariant differential forms, which can in turn 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.

Definition

Let be a Lie algebra over a commutative ring R with universal enveloping algebra

Universal enveloping algebra

In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...

 , and let M be a representation of (equivalently, a -module). Considering R as a trivial representation of , one defines the cohomology groups

(see Ext functor

Ext functor

In mathematics, the Ext functors of homological algebra are derived functors of Hom functors. They were first used in algebraic topology, but are common in many areas of mathematics.- Definition and computation :...

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

Analogously, one can define Lie algebra homology as

(see Tor functor

Tor functor

In homological algebra, the Tor functors are the derived functors of the tensor product functor. They were first defined in generality to express the Künneth theorem and universal coefficient theorem in algebraic topology....

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

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

Weyl's theorem

In mathematics, Weyl's theorem or Weyl's lemma might refer to one of a number of results of Hermann Weyl. These include* the Peter–Weyl theorem...

, and the Levi decomposition

Levi decomposition

In Lie theory and representation theory, the Levi decomposition, conjectured by Killing and Cartan and proved by , states that any finite dimensional real Lie algebra g is the semidirect product of a solvable ideal and a semisimple subalgebra....

theorem.

Cohomology in small dimensions

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

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

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

and is called inner if it is given by

for some a in M.

The second cohomology group

is the space of equivalence classes of Lie algebra extensions

of the Lie algebra by the module M.

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

External links