Lie algebra cohomology

In mathematics, the Lie algebra cohomology is a technical aid, which has particular application in differential geometry, mathematical physics and the theory of Lie groups. It is defined as the cohomology Koszul complex. For compact Lie groups which are algebraically defined Lie algebra cohomology of the Lie algebra is isomorphic to the de Rham cohomology of the Lie group.

Definition

Let be a Lie algebra. On the exterior algebra of the dual vector space, we define an operator for all

Be

Then we define

By

The complex is called Koszul complex. For all

The Lie algebra cohomology of the Koszul cohomology is defined as complex, so as

Lie groups and Lie algebras cohomology

For a Lie group with Lie algebra of the Koszul complex is canonically isomorphic to the complex of invariant differential forms:

The Lie algebras of - Komologie is isomorphic to the cohomology of the complex.

Elie Cartan has proved that for compact Lie groups, the inclusion

An isomorphism of the de Rham cohomology groups induced. So For compact Lie groups