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Killing form for gl_n

  1. Dec 2, 2007 #1
    1. The problem statement, all variables and given/known data
    Does anybody know of a concise way to prove that the Killing form of gl_n is given by [tex] B(X,Y) = 2n Tr(X \cdot Y) - 2 Tr(X) Tr(Y)[/tex]?

    2. Relevant equations

    3. The attempt at a solution

    I believe that this can be shown by noting that a basis for gl_n is given by the matrices E_{ij} whose only nonzero entries occur in the (i,j) entries. Using this fact, one could then explicitly work out the adjoint representation. Is there a simpler way, however?
  2. jcsd
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