# Lie subalgebra

1. Sep 24, 2011

### Ted123

1. The problem statement, all variables and given/known data

Let $\mathfrak{g}$ be the vector subspace in the general linear lie algebra $\mathfrak{gl}_4 \mathbb{C}$ consisting of all block matrices $$A=\begin{bmatrix} U& W\\ 0 & V\end{bmatrix}$$ where $U,V$ are any 2x2 matrices of trace 0 and $W$ is any 2x2 matrix.

Show that $\mathfrak{g}$ is a lie subalgebra in $\mathfrak{gl}_4 \mathbb{C}$.

2. Relevant equations

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

3. The attempt at a solution

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

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

$= \begin{bmatrix} UX & UZ+WY \\ 0 & VY \end{bmatrix} - \begin{bmatrix} XU & XW+ZV \\ 0 & YV \end{bmatrix}$

$= \begin{bmatrix} UX - XU & XW+ZV - WX- VZ\\ 0 & VY - YV \end{bmatrix}$

How can I show $[A,B]\in\mathfrak{g}$ ?

Last edited: Sep 24, 2011