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Help with Lie Algebra

  1. Jul 8, 2012 #1
    1. The problem statement, all variables and given/known data
    I have the Casimir second order operator:
    C= Ʃ gij aiaj
    and the Lie Algebra for the bases a:
    [as,al]= fpsl ap
    where f are the structure factors.

    I need to show that C commutes with all a, so that:
    [C,ar]=0


    2. Relevant equations
    gij = Ʃ fkilfljk

    (Jacobi identity for f is known, as well as it's antisymmetry to the lower indices)

    3. The attempt at a solution

    Well I go and write:
    (I am using Einstein's notation so that I won't keep the Sum signs, same indices are being added)
    [C,ar]=gij [aiaj,ar]
    =gij { ai [aj,ar] + [ai,ar ] aj }
    =gij { fpjr aiap + fpir apaj }

    Here starts my problem:
    I can't show that the above is zero... Any idea?
     
  2. jcsd
  3. Jul 9, 2012 #2
    Hmm...I have usually seen the Casimir operators defined as being elements of the centre of the universal enveloping algebra and then defining the structure constants, but I guess we can go the other direction.

    Without doing all the calculations, there are still a few more lines that we could add here that might lead to something fruitful. In particular, it seems to be that [itex] g_{ij} [/itex] is symmetric via its definition in terms of the structure constants. Hence [itex] g_{ij} f^p_{ir} a_p a_j = g_{ij} f^p_{jr} a_p a_i [/itex], unless of course I've completely forgotten how to symbol push. Then you can factor out to get [itex] g_{ij} f^p_{jr} (a_i a_p + a_p a_i) [/itex] and introduce a commutator here to one of the components. Hopefully magic happens and things cancel, though I'm honestly not certain.
     
  4. Jul 9, 2012 #3
    I have been thinking on it ( now it is just out of curiousity, since the deadtime is over).

    I agree with whay you've written and I used it reaching the same result but I can't make a solution out of it.

    If it was fijp it would be easier (i would have one symmetric*antisymmetric and I would get zero).
    Keep moving from your last result:

    gij fjrp (2aiap+ fpis as)

    I just used the Lie Algebra equation.

    2gij fjrp aiap +gij fjrp fpis as

    Changing the sumed indices p-->k for the first term, s-->k for the second

    2gij fjrk aiak +gij fjrp fpik ak

    So ak can come out:

    gij (2 fjrk ai + fjrp fpik ) ak

    That's where I reached it....
     
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