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

Famous quotes containing the words small and/or dimensions:

    There the wicked cease from troubling, and there the weary are at rest. There the prisoners are at ease together; they do not hear the voice of the taskmaster. The small and the great are there, and the slaves are free from their masters.
    Bible: Hebrew, Job 3:17-19.

    It seems to me that we do not know nearly enough about ourselves; that we do not often enough wonder if our lives, or some events and times in our lives, may not be analogues or metaphors or echoes of evolvements and happenings going on in other people?—or animals?—even forests or oceans or rocks?—in this world of ours or, even, in worlds or dimensions elsewhere.
    Doris Lessing (b. 1919)