Note: I’ve started putting together the material from these postings into a proper document, available here, which will be getting updated as time goes on. I’ll be making changes and additions to the text there, not on the blog postings. For most purposes, that will be what people interested in this subject will want to take a look at.
When a Lie group with Lie algebra acts on a manifold , one gets two sorts of actions of on the differential forms ). For each one has operators:
These operators satisfy the relation
where is the de Rham differential , and the operators are (super)-derivations. In general, an algebra carrying an action by operators satisfying the same relations satisfied by will be called a -differential algebra. It will turn out that the Clifford algebra of a semi-simple Lie algebra carries not just the Clifford algebra structure, but the additional structure of a -differential algebra, in this case with , not grading.
Note that in this section the commutator symbol will be the supercommutator in the Clifford algebra (commutator or anti-commutator, depending on the grading). When the Lie bracket is needed, it will be denoted .
To get a -differential algebra on we need to construct super-derivations , , and satisfying the appropriate relations. For the first of these we don’t need the fact that this is the Clifford algebra of a Lie algebra, and can just define
For , we need to use the fact that since the adjoint representation preserves the inner product, it gives a homomorphism
where is the Lie algebra of the group (the spin group for the inner product space ), which can be identified with quadratic elements of , taking the commutator as Lie bracket. Explicitly, if is a basis of , the dual basis, then
and we get operators acting on
Remarkably, an appropriate can be constructed using a cubic element of . Let
since is a scalar which can be computed to be , where is the Casimir operator in the adjoint representation.
The above constructions give the structure of a filtered -differential algebra, with associated graded algebra . This gives the structure of a -differential algebra, with operators . The cohomology of this differential algebra is just the Lie algebra cohomology .
can be thought of as an algebra of operators corresponding to the quantization of an anti-commuting phase space . Classical observables are anti-commuting functions, elements of . Corresponding to one has both elements of and their quantizations, the operators in constructed above.
For more details about the above, see the following references