Construction of base form
Let
denote the
equivariant form, given
by Eq. (
30). We will denote the
corresponding base form
:
We will work under the assumption that:
Therefore
can be considered
a principal
-bundle.
In order to construct the base form
from the Cartan’s
, we first choose on
this principal bundle
some connection
. (We understand the connection as a
-valued 1-form
on
computing the “vertical component” of a vector.) Then we apply
the horizonthal projection
i.e. replace
Finally, we replace
with
the curvature
of the connection
; we get:
To conclude:
Generally speaking, it is
not true that
descends from
to
. But we found a class of subalgebras of
, of the form
,
such that for any subalgebra from this class we can construct an equivariant form
. Then the standard procedure can be used to construct
the corresponding base form
which descends to
.
The result, generally speaking, does depend on the choice of
.
