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
Famous quotes containing the words small and/or dimensions:
“It doesn’t matter that your painting is small. Kopecks are also small, but when a lot are put together they make a ruble. Each painting displayed in a gallery and each good book that makes it into a library, no matter how small they may be, serves a great cause: accretion of the national wealth.”
—Anton Pavlovich Chekhov (1860–1904)
“I was surprised by Joe’s asking me how far it was to the Moosehorn. He was pretty well acquainted with this stream, but he had noticed that I was curious about distances, and had several maps. He and Indians generally, with whom I have talked, are not able to describe dimensions or distances in our measures with any accuracy. He could tell, perhaps, at what time we should arrive, but not how far it was.”
—Henry David Thoreau (1817–1862)