# Eigenvalues of quadratic Casimirs of simple Lie groups

I want to find a generic formula for calculating eigenvalue of quadratic casimirs of Lie groups, in terms of Dynkin labels.

For a simple example if we take $$SU(2)$$, with $$[R]$$ indicating the highest weight of the $$(R+1)$$-dim representation, I would get $$(R/2)(R/2+1)$$.

Is there a formula for other series of classical Lie algebras as well?

Yes. Up to normalization one finds, for $$\hat e_k$$ in a Hermitian basis of the algebra, $$\hat C_2=\frac{1}{2}\sum_{k=1}^\ell \hat e_k^2$$ with eigenvalue $$c_2=2\langle \Lambda\vert \delta\rangle +\langle \Lambda\vert\Lambda\rangle$$ where $$\vert\Lambda\rangle=\sum_{i=1}^r \lambda_i\vert w_i\rangle$$ is the highest weight state expressed in terms of the fundamental weights $$\{\vert w_i\rangle\}$$ and $$\delta$$ is the Weyl root, which is half of the sum of all positive roots.

The details on the derivation of this result can be found in

J. F. Cornwell, Group Theory in Physics (Academic, New York, 1984), Vol. 2.

and worked out examples can be found in

R. Slansky, Phys. Rep. 79, 1 (1981)

or

Iachello, F., 2006. Lie algebras and applications (Vol. 12). Berlin: Springer

Yes. For any simple Lie algebra, if $$\lambda$$ is the highest weight in a representation and $$\rho$$ is the Weyl vector (the sum of fundamental weights or 1/2 the sum of the positive roots) then the Casimir is given by $$Q= (\lambda, \lambda+2\rho)$$. Here the inner product is the standard one on the weight space.

I regard Chapter 13 in Di Francesco, Mathieu and Senenschal's "Conformal Field theory" to be the best refernce on Lie algebra stuff for physicists. It's a completely self-contained acount of nearly everything one needs to know. You don't need to know an Conformal Field theory to read that chapter. It has a clear proof of this result.