1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    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!

Trace((ad_x)(ad_y)) = 2n(trace(xy))

  1. Apr 25, 2010 #1
    1. The problem statement, all variables and given/known data

    Let L be the Lie algebra [tex]sl(n, F)[/tex] and [tex]X = (x_{ij}, Y = (y_{ij}) \in L[/tex].


    [tex]\kappa(X,Y) = 2n Tr(XY)[/tex],

    where [tex]\kappa(,)[/tex] is the Killing form and [tex]Tr()[/tex] is the trace form.

    2. Relevant equations

    For any unit matrix [tex]E_{ij}[/tex] and any [tex]X \in L[/tex],

    [tex]XE_{ij} = \sum_{m=1}^n x_{mi} E_{mj}[/tex] and
    [tex]E_{ij}X = \sum_{m=1}^n x_{jm}E_{im}.[/tex]

    3. The attempt at a solution

    I have reduced this to the following:

    [tex]Tr(XY) = \sum_{k=1}^n \sum_{m=1}^n x_{mk}y_{km}[/tex]

    [tex]\kappa(X,Y) = Tr(ad_X ad_Y) = \sum_{k=1}^n (x_{ik}y_{ki} + y_{kj}x_{jk} ) - 2 \sum_{i=1}^n \sum_{j=1}^n x_{ii}y_{jj}[/tex]

    Perhaps I have been at this for too long, but I don't see why exactly 2n times the first expression is equivalent to the second expression. Any guidance would be appreciated.
    Last edited: Apr 26, 2010
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

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

Similar Threads - Trace ad_x ad_y Date
Trace of six gamma matrices Nov 4, 2015
Trace of a particular matrix product Apr 10, 2015
Simple indice question -- metric, traces Dec 18, 2014