I’m studying representation theory of semisimple Lie algebras through highest weights. We have seen that finite dimensional irreducible representations are in 1 to 1 correspondence with dominant weights. In the case of $sl(2;\mathbb{C})$ we have only one simple root $\alpha_1$, with fundamental dominant weight $w_1=\frac{\alpha_1}{2}$. Thus, finite dimensional irreducible representations of $sl(2;\mathbb{C})$ are isomorphic to highest weight representations $M_\lambda$ with $\lambda=w_1,2w_1,3w_1,...$, or what is the same, with $\lambda=\frac{\alpha_1}{2},\alpha_1,\frac{3\alpha_1}{2},...$. Now, this reminds one to spin representations in physics (which are irreps of $sl(2;\mathbb{C})$). But what does the simple root $\alpha_1$ has to do with eigenvalues of $J^2$?
Also, the set of weights of $M_\lambda$, with $\lambda=\frac{m\alpha_1}{2}$ is: $P(M_\lambda)={\frac{m\alpha_1}{2}},\frac{m-2\alpha_1}{2},...,0,...,\frac{-m-2\alpha_1}{2},\frac{-m\alpha_1}{2}$, which suspiciously resembles the posible values of the 3rd component of Spin, $J_3$. Why?
So my question is, what’s going on? I feel that I’m almost there, finally understanding the irreducible representations of the spin that physicist introduce so easily by the casimir element $J^2$ and ladder operators. But I’m lacking some relation to this weight representation theory of Lie algebras.
Note: I haven’t seen any theory related to representations through casimir operators, and I think the key is here. So I would infinitely appreciate an answer which builds from what I know and makes the explicit relations that I’m lacking.