# Why the Casimir operator is proportional to the unit matrix ?

1. Feb 22, 2013

### Wonchu

Hello,
I have a question about the Casimir operator on page 500 in Chapter 15.

From the following eq,
$\ \ \ [T^b , T^a T^a ] = 0 \ \ \ \ \ \ \ (15.91)$
$T^2(=T^a T^a)$ is an invariant of the algebra.
Thus the author concludes that $T^2$ is proportional to the unit matrix.
Why is that ?
How to prove it ?

2. Feb 22, 2013

### kevinferreira

That is called Schur's lemma. Look for it. If an element of the algebra commutes with every element it must be the proportional to the identity (as this is the unique element which has this property).

3. Feb 22, 2013

### Wonchu

Now that you say that,
I recollect I also have heard about that lemma.
Now I can relate to.

Thanks a lot !