What is the simplicity of the Special Linear Lie Algebra?

[heras1985]

Hi,
Show that the Special linear Lie algebra is simple.
I tried it with induction but without result.

 

[Pere Callahan]

Welcome to PF.

What exactly have tried? A good point to start is citing the definition of "Special linear Lie algebra" and "simple".

 

[heras1985]

Definition of simple:
L is called simple if it has no ideals except {0} and L.
I is an ideal of L if $$x\in L, y\in I \Rightarrow [x,y]\in I$$
The matrices whose trace is 0 form the special linear lie algebra $$sl_n(\mathbb{C})$$. The special linear lie algebra is the lie algebra of the special linear group (this group is form by the matrices whose determinant is 1).