Integration measures from representations of and

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    PDFs from representations of
        Mapping to cochains
        Mapping to PDFs
    PDFs from representations of

PDFs from representations of

If acts on a manifold, then acts in PDFs. More generally, acts in cochains of Chevalley-Eilenberg complexes of .

Question: Given some representation of , can we map it to PDFs, or to cochains?

Mapping to cochains

Proposition 2. Let

be a

-module, and

an intertwiner of

-modules:

satisfying:

(26)

(One may think of

as an “integration operation”.) Consider the subspace

Then operation

intertwines this subspace with the Chevalley-Eilenberg complex

Proof follows from Eq. (25).

Therefore every

defines an inhomogeneous Chevalley-Eilenberg cochain of

with coefficients in

The map

intertwines the action of

with the standard action of

in cochains with coefficients in

— the

of Proposition 1. (This action does not use

, generally speaking there is no such thing as

. Our

, unlike

, is just a

-module, not a differential

-module.)

Mapping to PDFs

Suppose that

happens to be a space of functions on some manifold

with an action of

(the Lie group corresponding to

). In this case, every

and a point

defines a closed PDF on

, in the following way:

(27)

We will be mostly interested in the cases when this PDF descends to the

-orbit of

For example, consider the case when is the space of PDFs on (the same ) and is the restriction of a PDF on the zero section . (Remember that PDFs are functions on . In this example, the operation associates to every form its 0-form component.) In this case, given a PDF on , e.g. , our procedure, for each , associates to it a PDF on , which is just . If acts freely, will descend to a form on the orbit of . This is just the restriction to the orbit of the original form we started with.

As another example, consider the Lie algebra of vector fields on some manifold , and the space of PDFs on . Let be the space of orientable -dimensional submanifolds of , and the operation of integration over such a submanifold. Our construction maps closed forms on to closed forms on .