Editing Representation theory of SU(2) (section)

===The Casimir element===
We now introduce the (quadratic) [[Casimir element]], <math>C</math> given by
:<math>C = -\left(u_1^2 + u_2^2 + u_3^2\right)</math>.

We can view <math>C</math> as an element of the [[universal enveloping algebra]] or as an operator in each irreducible representation. Viewing <math> C </math> as an operator on the representation with highest weight <math> m </math>, we may easily compute that <math> C </math> commutes with each <math> u_i  .</math> Thus, by [[Schur's lemma]], <math> C </math> acts as a scalar multiple <math>c_m</math> of the identity for each <math> m .</math> We can write <math>C</math> in terms of the <math>\{ H, X, Y \} </math> basis as follows:
:<math>C = (X + Y)^2 - (-X + Y)^2 + H^2  ,</math>
which can be reduced to
:<math>C = 4YX + H^2 + 2H .</math>

The eigenvalue of <math> C </math> in the representation with highest weight <math> m </math> can be computed by applying <math> C </math> to the highest weight vector, which is annihilated by <math> X  ;</math> thus, we get 
:<math>c_m = m^2 + 2m = m(m + 2) .</math>

In the physics literature, the Casimir is normalized as <math display="inline"> C' = \frac{1}{4}C .</math> Labeling things in terms of <math display="inline"> \ell = \frac{1}{2}m  ,</math> the eigenvalue <math> d_\ell </math> of <math> C' </math> is then computed as
:<math> d_\ell = \frac{1}{4}(2\ell)(2\ell + 2) = \ell (\ell + 1) .</math>