Casimir element

Summary

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra with a nondegenerate symmetric invariant bilinear form. Its Casimir element is a quadratic element of the centre of the universal enveloping algebra $U(\mathfrak{g})$. It acts by a scalar on each irreducible finite-dimensional module of a complex semisimple Lie algebra. For the Killing form, this scalar is nonzero on every nontrivial irreducible module. This makes the Casimir element a key tool in the proof of Weyl's theorem on complete reducibility. In physics, the Casimir of the rotation algebra is the total angular momentum squared $J^2$.