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General linear lie algebra

  1. Sep 24, 2011 #1
    1. The problem statement, all variables and given/known data

    Let [itex]\mathfrak{g}[/itex] be the vector subspace in the general linear lie algebra [itex]\mathfrak{gl}_4 \mathbb{C}[/itex] consisting of all block matrices [tex]A=\begin{bmatrix} X & Z\\ 0 & Y \end{bmatrix}[/tex] where [itex]X,Y[/itex] are any 2x2 matrices of trace 0 and [itex]Z[/itex] is any 2x2 matrix.

    You are given that [itex]\mathfrak{g}[/itex] is a lie subalgebra in [itex]\mathfrak{gl}_4 \mathbb{C}[/itex].

    Consider [itex]\mathfrak{g}[/itex] as a lie algebra.

    Prove that the radical of [itex]\mathfrak{g}[/itex] consists of all matrices [itex]A[/itex] where [itex]X=Y=0[/itex].

    You may use the fact that the lie algebra [itex]\mathfrak{sl}_2 \mathbb{C}[/itex] which consists of all 2x2 matrices of trace 0 is simple.

    3. The attempt at a solution

    How would I go about this?
     
  2. jcsd
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