On this page:
1 Notations
2 Central extension
3 Moment map

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 :

(23)

Let 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:

(24)

(25)

(26)

where:
  • the action of on is induced from the following action on :

  • the action of on is: