Calculating Moment Map of 2-Torus Action on C^3

[HMY]

Homework Statement

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?

Homework Equations

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.

 

[mighty2000]

Thank you for your post. It seems like you have made a good attempt at computing the moment map for the given 2-torus action on C^3. However, there are a few mistakes in your calculations that I would like to point out.

Firstly, the Lie algebra for this 2-torus action is not isomorphic to the dual Lie algebra. The Lie algebra for this action is actually isomorphic to the Lie algebra of the 2-torus itself, which is given by the set of all skew-symmetric matrices with real entries. This Lie algebra is not dual to R^2.

Secondly, your calculations for the vector fields that generate the action are not correct. The correct vector fields are:

1) partial wrt theta_1
2) partial wrt theta_2
3) partial wrt theta_3

These vector fields can be obtained by taking the tangent vectors to the orbits of the 2-torus action. Your calculations seem to be based on a different action, which is not the same as the one given in the problem.

Finally, your moment map is not correct. The moment map for this 2-torus action is given by:

u(x, y, z) = (|x|^2/2, |y|^2/2, |z|^2/2)

This can be obtained by using the definition of the moment map and the vector fields given above.

I hope this helps clarify your doubts and helps you in finding the correct solution. Good luck with your further calculations!