Lie algebra cohomology

In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was defined by Chevalley and 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.

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. 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.

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

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