Descent to double coset
The form
does not descend from
to
; but we will prove that for some subgroups
, it can be modified in a natural way, so that the modified form does descend to
1 Straightforward descent does not work
We are identifying 
with the
double coset 
.
Let 
be a special
canonical transformation, i.e. 
. It follows immediately
that 
is invariant under
the left shift 
. But
is not horizonthal; for 
we have:
with the
double coset .
Let be a special
canonical transformation, i.e. . It follows immediately
that is invariant under
the left shift . But
is not horizonthal; for we have:
|
Therefore:
to
2 Modified PDF
However, we identify a class of subalgebras 
for which we can construct
the base form 
. Being a base form, it descends to 
,
where 
is the Lie group generated by the flows of elements of 
.