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Structure constants of Lie groups

  1. Apr 15, 2004 #1


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    Source: Anderson, Principles of Relativity Physics

    p. 13, prob. 1.4

    "Reparametrize the rotation group by taking, as new infinitesimal parameters, ε1 = ε23, ε2 = ε31, and ε3 = ε12 and calculate the structure constants for these parameters."

    My assumptions:

    The εij mentioned in the problem are the infinitesimal Cartesian parameters of the 3-D rotation group such that εij = -εji, and yi = xi + Σjεijxj, where x is the original point and y is the transformed point.

    To generalize this to non-Cartesian coordinates and still maintain the Lie group-ness, the transformation takes the general form:

    yi = xi + Σkεkfki(x)

    where the fki(x) satisfy the following condition.

    The request for structure constants is a request for constants ckmn such that:

    yi = xi + ΣkΣmΣnBmεAn - εAmεBn)ckmnfki(x)

    The parameters εk are the non-Cartesian parameters, and so, they should multiply some functions fki(x), and these functions determine the structure constants.

    My problem with understanding:

    I don't know how to find the fki(x). I have:

    Σjεijxj = Σkεkfki(x)

    but I don't see how this tells me fik(x). Am I supposed to assume some kind of orthogonality or something?
    Last edited: Apr 15, 2004
  2. jcsd
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