Odd symplectic manifolds
1 Contraction and Lie derivative
We define
for a vector field
as follows. If
is even, we pick
a Grassmann odd parameter
and define:
If
is odd, we define
as follows:
.
In coordinates:
The relation to Lie derivative:
2 Symplectic structure and Poisson structure
Consider a supermanifold
, with local coordinates
, equipped with an
odd Poisson bracket
of the form:
The Poisson bivector
should be symmetric in the following sense:
The odd symplectic form
can be
defined from the following equation:
3 Darboux coordinates
In Darboux coordinates:
parametrizes a point in the fiber of
over the point
in
. Then
is a new point in the same fiber, linearly depending on
.
