In this posting I’ll work out some examples of Lie algebra cohomology, still for finite dimensional Lie algebras and representations.
If is a compact, connected Lie group, it can be thought of as a compact manifold, and as such one can define its de Rham cohomology as the cohomology of the complex
where are the differential i-forms on (note, we’ll use complex-valued forms), and is the deRham differential.
For a compact group, one has a bi-invariant Haar measure , and can use this to “average” over an action of the group on a space. For a representation , we get a projection operator onto the invariant subspace . This projection operator gives explicitly the invariants functor on . It is an exact functor, taking exact sequences to exact sequences.
The differential forms give a representation of in two ways, taking the induced action on forms by pullback, using either left or right translation on the group. If is the representation by left translations, we can use this to apply our “averaging over ” projection operator to the de Rham complex. This action commutes with the de Rham differential, so we get a sub-complex of left-invariant forms
Since elements of the Lie algebra are precisely left-invariant 1-forms, it turns out that this complex is nothing but the Chevalley-Eilenberg complex considered last time to represent Lie algebra cohomology, for the case of the trivial representation. This means we have , and the differentials coincide. So, what we have shown is that
If one knows the cohomology of , the Lie algebra cohomology is thus known, but this identity is normally used in the other direction, to find the cohomology of from that of the Lie algebra. To compute the Lie-algebra cohomology, we can exploit the right-action of G on the group, averaging over the induced action on the left-invariant forms , which again commutes with the differential. We end up with a complex
where all the differentials are zero, so the cohomology is given by
the adjoint-invariant pieces of the exterior algebra on . Finding the cohomology has now been turned into a purely algebraic problem in invariant theory. For , , and we have shown that , this is in degrees 0, and 1, as expected for the de Rham cohomology of the circle . For , we get
Note that complexifying the Lie algebra and working with commutes with taking cohomology, so we get
Complexifying the Lie algebra of a compact semi-simple Lie group gives a complex semi-simple Lie algebra, and we have now computed the cohomology of these as
Besides , one always gets a non-trivial , since one can use the Killing form to produce an adjoint-invariant 3-form . For , , and one gets non-trivial cohomology classes for , such that
the exterior algebra generated by the .
To compute Lie algebra cohomology with coefficients in a representation , we can go through the same procedure as above, starting with differential forms on taking values in , or we can just use exactness of the averaging functor that takes to . Either way, we end up with the result
The piece of this is just the that we want when we are doing BRST, but we also get quite a bit else: copies of the higher degree pieces of the Lie algebra cohomology . The Lie algebra cohomology here is quite non-trivial, but doesn’t interact in a non-trivial way with the process of identifying the invariants in .
In the next posting I’ll turn to an example where Lie algebra cohomology interacts in a much more interesting way with the representation theory, this will be the highest-weight theory of representations, in a cohomological interpretation first studied by Bott and Kostant.