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Lie Algebra's question

  1. Apr 12, 2007 #1
    Take L to be a subspace of sl (2,R). (R is the real numbers)

    [tex]
    L = \left(
    \begin{array}{ccc}
    0 & -c & b\\
    c & 0 & -a\\
    -b & a & 0
    \end{array}\right)
    [/tex]

    The basis elements of L are

    [tex]
    e_1 = \left(
    \begin{array}{ccc}
    0 & 0 & 0\\
    0 & 0 & -1\\
    0 & 1 & 0
    \end{array}\right)
    [/tex]

    [tex]
    e_2 = \left(
    \begin{array}{ccc}
    0 & 0 & 1\\
    0 & 0 & 0\\
    -1 & 0 & 0
    \end{array}\right)
    [/tex]

    [tex]
    e_3 = \left(
    \begin{array}{ccc}
    0 & -1 & 0\\
    1 & 0 & 0\\
    0 & 0 & 0
    \end{array}\right)
    [/tex]

    What is the best way to show that this Lie Algebra is simple (i.e. the only ideals are {0} and L? I know it's non-abelian.
     
  2. jcsd
  3. Apr 12, 2007 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    By working out the [ ] of e_i and e_j, you should see how to show that the commutator spans the whole of the L again. That should do it.

    Or you can just write down an isomorphism to sl_2(R). (Surely you meant to say L is a subspace of sl_3).
     
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