Showing \mathfrak{g} is a Lie Subalgebra in \mathfrak{gl}_4 \mathbb{C}

[Ted123]

Homework Statement

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 $\mathfrak{gl}_4 \mathbb{C}$.

Homework 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).

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 $[A,B]\in\mathfrak{g}$ ?

 

[mighty2000]

To show that [A,B]\in\mathfrak{g}, we need to show that the resulting matrix is in the form of A=\begin{bmatrix} U & W \\ 0 & V \end{bmatrix}, where U and V are 2x2 matrices of trace 0 and W is a 2x2 matrix.

From the previous calculations, we can see that the top left block of [A,B] is the difference of two 2x2 matrices, which is still a 2x2 matrix. Similarly, the top right block is also a 2x2 matrix.

For the bottom left block, we have 0 as the top row and the difference of two 2x2 matrices as the bottom row. This still gives us a 2x2 matrix.

Lastly, the bottom right block is the difference of two 2x2 matrices, which is still a 2x2 matrix.

Therefore, we can conclude that [A,B]\in\mathfrak{g} and \mathfrak{g} is a lie subalgebra of \mathfrak{gl}_4 \mathbb{C}.