(adsbygoogle = window.adsbygoogle || []).push({}); 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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: General linear lie algebra

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**