# Help with Lie Algebra

1. Jul 8, 2012

### Morgoth

1. The problem statement, all variables and given/known data
I have the Casimir second order operator:
C= Ʃ gij aiaj
and the Lie Algebra for the bases a:
[as,al]= fpsl ap
where f are the structure factors.

I need to show that C commutes with all a, so that:
[C,ar]=0

2. Relevant equations
gij = Ʃ fkilfljk

(Jacobi identity for f is known, as well as it's antisymmetry to the lower indices)

3. The attempt at a solution

Well I go and write:
(I am using Einstein's notation so that I won't keep the Sum signs, same indices are being added)
[C,ar]=gij [aiaj,ar]
=gij { ai [aj,ar] + [ai,ar ] aj }
=gij { fpjr aiap + fpir apaj }

Here starts my problem:
I can't show that the above is zero... Any idea?

2. Jul 9, 2012

### Kreizhn

Hmm...I have usually seen the Casimir operators defined as being elements of the centre of the universal enveloping algebra and then defining the structure constants, but I guess we can go the other direction.

Without doing all the calculations, there are still a few more lines that we could add here that might lead to something fruitful. In particular, it seems to be that $g_{ij}$ is symmetric via its definition in terms of the structure constants. Hence $g_{ij} f^p_{ir} a_p a_j = g_{ij} f^p_{jr} a_p a_i$, unless of course I've completely forgotten how to symbol push. Then you can factor out to get $g_{ij} f^p_{jr} (a_i a_p + a_p a_i)$ and introduce a commutator here to one of the components. Hopefully magic happens and things cancel, though I'm honestly not certain.

3. Jul 9, 2012

### Morgoth

I have been thinking on it ( now it is just out of curiousity, since the deadtime is over).

I agree with whay you've written and I used it reaching the same result but I can't make a solution out of it.

If it was fijp it would be easier (i would have one symmetric*antisymmetric and I would get zero).
Keep moving from your last result:

gij fjrp (2aiap+ fpis as)

I just used the Lie Algebra equation.

2gij fjrp aiap +gij fjrp fpis as

Changing the sumed indices p-->k for the first term, s-->k for the second

2gij fjrk aiak +gij fjrp fpik ak

So ak can come out:

gij (2 fjrk ai + fjrp fpik ) ak

That's where I reached it....