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Lie subalgebra

  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} U& W\\ 0 & V\end{bmatrix}[/tex] where [itex]U,V[/itex] are any 2x2 matrices of trace 0 and [itex]W[/itex] is any 2x2 matrix.

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

    2. Relevant equations

    A subspace [itex]\mathfrak{g}[/itex] of [itex]\mathfrak{gl}_4 \mathbb{C}[/itex] is a lie subalgebra of [itex]\mathfrak{gl}_4 \mathbb{C}[/itex] if for all [itex]x,y\in\mathfrak{g}[/itex] it follows that [itex][x,y]\in\mathfrak{g}[/itex] where [itex][\cdot , \cdot ][/itex] is the lie bracket in [itex]\mathfrak{gl}_4 \mathbb{C}[/itex] (the matrix commutator: [X,Y]=XY-YX).

    3. The attempt at a solution

    Let [itex]A=\begin{bmatrix} U & W \\ 0 & V \end{bmatrix},B=\begin{bmatrix} X & Z \\ 0 & Y \end{bmatrix}\in\mathfrak{g}[/itex]

    Then [itex][A,B]=AB-BA=\begin{bmatrix} U & W \\ 0 & V \end{bmatrix} \begin{bmatrix} X & Z \\ 0 & Y \end{bmatrix} - \begin{bmatrix} X & Z \\ 0 & Y \end{bmatrix} \begin{bmatrix} U & W \\ 0 & V \end{bmatrix}[/itex].

    [itex]= \begin{bmatrix} UX & UZ+WY \\ 0 & VY \end{bmatrix} - \begin{bmatrix} XU & XW+ZV \\ 0 & YV \end{bmatrix}[/itex]

    [itex]= \begin{bmatrix} UX - XU & XW+ZV - WX- VZ\\ 0 & VY - YV \end{bmatrix}[/itex]

    How can I show [itex][A,B]\in\mathfrak{g}[/itex] ?
     
    Last edited: Sep 24, 2011
  2. jcsd
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