# G* Modules
# From Geometry to Algebra
Let be any Lie group, let be its Lie algebra, and be the corresponding Lie superalgebra.
A algebra is a commutative superalgebra , together with a representation of as automorphisms of and an action of as (super)derivations of which are consistent in the sense that
for all , .
A module is a supervector space together with a linear representation of on and a homomorphism such that the conditions above hold. So a algebra is a commutative superalgebra which is a module with the additional condition that acts as algebra automorphisms and acts as superderivations.
In order to make the set of modules and the set of algebras into category, we must first define what is meant by a morphism. So let and be modules and let be a (continuous) linear map.
We say that is a morphism of modules if for all , , , we have
If, for all ,
we say that has degree , with similar notation for the -graded case. We say that a morphism of degree is even if and odd if .
If the morphism is even (especially if it is of degree zero which will often be the case), one can say that preserves the action, or that the operators , and are equivariant with respect to .
The composition of two morphisms is again a module morphism, and hence the set of module morphisms is a category. Further, we define a morphism between algebras to be a map which is an algebra morphism, and satisfies the four equations above. This makes the set of algebras into a category too. Similar analogous definitions exist of -graded modules, algebras, and morphisms.
If we have a -action on a manifold , then is a algebra in a canonical way. If and are -manifolds and is a -equivariant smooth map, then the pullback map is a morphism of algebras. Thus the category of algebras can be considered as an algebraic generalisation of the category of -manifolds.
By definition, the element acts as a derivation of degree with on . So is a cochain complex. We define to be the cohomology of relative to the differential . In the case when , de Rham's theorem says that this is equal to .
(to be added - page 19)
If is contractible, the de Rham complex is acyclic, i.e., satisfies
where is the ground field, which is in our case. We take this as the definition of acyclicity for a general .
# Chain Homotopies
Let and be two modules. A linear map
is called a chain homotopy if it is odd, -equivariant, and satisfies
If and are -graded (as is typical) we require that be of degree in the -gradation. The -equivariance implies that
If is a chain homotopy, then
is a morphism of modules.
We say that two morphisms are chain homotopic (notation: ) if there is a chain homotopy such that
This implies that the induced maps on cohomology are equal:
(finish chain homotopy relative to... - page 22).
# Free and Locally Free Actions
An action of on is said to be locally free if the corresponding infinitesimal action of is free, i.e., if, for every , the vector field generating the one-parameter group of transformations on is nowhere vanishing.
If the action is locally free, we can find linear differential forms on which are everywhere dial to our basis in the sense that
Conversely, if we have a -action on a manifold on which there exist forms satisfying the above equation, then the action is locally free.
A linear differential form is called horizontal if it satisfies