For a Lie group
and a vector
bundle
we study those
actions of the Lie group
on
for which
the action map
is a morphism of vector bundles, and call those
affine actions. We prove that the category
of such actions over
a fixed
-manifold
is equivalent to a
certain slice category
.
We show that there is a monadic adjunction relating
to
, and
the right adjoint of this adjunction induces an isomorphism of Grothendieck groups
.
Complexification produces analogous results involving
and
.
