# Casimir operator

Let $\mathfrak{g}$ be a semisimple Lie algebra, and let $(\cdot,\cdot)$ denote the Killing form. If $\{g_{i}\}$ is a basis of $\mathfrak{g}$, then there is a dual basis $\{g^{i}\}$ with respect to the Killing form, i.e., $(g_{i},g^{j})=\delta_{ij}$. Consider the element $\Omega=\sum g_{i}g^{i}$ of the universal enveloping algebra of $\mathfrak{g}$. This element, called the is central in the enveloping algebra, and thus commutes with the $\mathfrak{g}$ action on any representation.

Title Casimir operator CasimirOperator 2013-03-22 13:52:53 2013-03-22 13:52:53 bwebste (988) bwebste (988) 5 bwebste (988) Definition msc 17B20