Integration measures from representations of and
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
Mapping to PDFs
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
.
PDFs from representations of