# Moment map

1. Apr 11, 2007

### HMY

1. The problem statement, all variables and given/known data

Given a 2-torus action on C^3 can be defined by
(a,b).(x,y,z)= (abx, a^-1by, bz). What is the moment
map?

2. Relevant equations

3. The attempt at a solution

Here the Lie algebra is isomorphic to the dual Lie algebra
which is dual to R^2.

I was trying to compute the moment map directly.

The vector fields that generate the action in polar
coordinates on C^3 are (I think):
1) partial w.r.t theta_1 - partial w.r.t theta_2
2) partial wrt theta_1 + partial wrt theta_2
+ partial wrt theta_3
3) partial wrt theta_3

Why do I think those: I'm first thinking of b=1 & letting a
vary, then for the other one I do the reverse, a=1 & b
varies.

Then do calculation of
-(interior derivative of "1)")omega
= ... (some work I'm leaving out for now)
= d( |x|^2 /2 - |y|^2 /2 )

Similarly for "2)" & "3)".

So end of day I think the moment map is:
u(x, y, z) =
(|x|^2/2 -|y|^2/2 , |x|^2/2 +|y|^2/2 +|z|^2/2, |z|^2/2 )

I also think I've done something wrong in this attempt.