Lie Algebra Cohomology - Cohomology in Small Dimensions

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.

Read more about this topic:  Lie Algebra Cohomology

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