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 Casimir operator is central in the enveloping algebra, and thus commutes with the $\mathfrak{g}$ action on any representation.

Title	Casimir operator
Canonical name	CasimirOperator
Date of creation	2013-03-22 13:52:53
Last modified on	2013-03-22 13:52:53
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	5
Author	bwebste (988)
Entry type	Definition
Classification	msc 17B20