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Roots and Weights in Lie algebras

  1. Mar 15, 2012 #1
    Hello! Im trying to understand this subject but its not simple..
    I will ask some question but if anybody wants to write a short introduction which explains my confusions in a continiuous text, that would be awesome as well. :)

    I think I got a good view of what our weigts are.. just vectors of our eigenvalues.
    And our roots are just the eigenvalues in the adjoint representation.

    When the author (Im currently reading Giorgi's book) takes this to SU(2) he talks about either using E_\alpha or E_\beta... but SU(2) just got one pair of creation- and destruction operators right? I think of the "E-operators" as creation and destruction but is there really anything to choose from in SU(2)?

    Futhermore, the simple positive roots are always a total number of m, because this are the number of cartan generators, i.e. the observable we have been able to diagonalize simultanious. So each simple positive root is a creation operator for an observable which can be measured at the same time as the other observables/operators in our cartan subalgebra?

    Lastly I´ve seen the masterformula and the derivation but when is it needed? Am I right if the main point is that if we have the dimension of our representation and one root, we can calculate all other roots? :/ Is there much more to it?



    Thanks really much!
    Best regards
    Hymne
     
  2. jcsd
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