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I need to construct the Casimir op. of group SU(3).

I have these relations;

T^{2}=[tex]\sum C_{i}_{j}T_{i}T_{j}[/tex] i,j=1,2....,8 ...eq1

[T_{i}, T_{j}]= [tex]\sum f_{i,j,k} T_{k}[/tex] ...eq2

[T^{2}, T_{i}]=[[tex]\sum C_{i}_{j}T_{i}T_{j}[/tex] , T_{s}]=[tex]\sum C_{i}_{j}T_{i}[T_{j}, T_{s}] + \sum C_{i}_{j}[T_{i}, T_{s}]T_{j} [/tex]=0 ...eq3

[T_{j},T_{s}]=[tex]\sum f_{j,s,m} T_{m}[/tex] ...eq4

[T_{i},T_{s}]=[tex]\sum f_{i,s,m} T_{m}[/tex] ...eq5

[T^{2}, T_{i}] = [tex]\sum C_{i}_{j} \sum f_{j,s,m} T_{i} T_{m} + \sum C_{i}_{j} \sum f_{i,s,m} T_{m}T_{j} [/tex]=0 ...eq6

by the way;

* I normalized the system. i=1

** i'm still improoving my mathematical skills because of that i might use more sum operator that i need. sorry for that.

*** All the indexes are in [1,8] range, i,j,k,s,m = 1,2,....,8

**** f coefficients are known!

I have the T_{i}matrices but i need to compute C_{i,j}constants to construct the quad. Casimir op.

1. Is the eq6 right?

2. If it's right , can i collect all of components in ONE sum operator? how the indexes change?

3. How could i compute the C_{i,j}constants?

Thanks for you patience and advices.

ToreHan.

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# Quadratic Casimir Operator of SU(3)

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