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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