Moment map
1 Notations
Let 
denote the group of canonical transformations of 
.
Let us denote 
and 
the group of left and right shifts on (both and are
naturally isomorphic to ). Both left and right shifts naturally lift to
. Let 
be the Lie algebra corresponding to .
2 Central extension
Let 
denote the space of functions on . Notice that 
is a
Lie superalgebra under the Poisson bracket. It is a central extension of :
denote the space of functions on . Notice that is a
Lie superalgebra under the Poisson bracket. It is a central extension of :
Let 
denote the corresponding Lie group.
denote the corresponding Lie group.3 Moment map
As usual, we consider the right-invariant differential form on with values
in 
, which is denoted 
. Because of (23), we can consider it as a
differential 1-form with values in . There is some invariance under
the left and right shifts, which can be summarized as follows:
with values
in , which is denoted . Because of (23), we can consider it as a
differential 1-form with values in . There is some invariance under
the left and right shifts, which can be summarized as follows:
where:
- the action of
on 
is induced from the
following action on :
- the action of on is:
