1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Structure constants of Lie groups

  1. Apr 15, 2004 #1


    User Avatar
    Homework Helper

    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted