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Moment map

  1. Apr 11, 2007 #1


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    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

    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

    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.
  2. jcsd
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