Casimir operator

Let 𝔤 be a semisimple Lie algebraMathworldPlanetmath, and let (,) denote the Killing formPlanetmathPlanetmath. If {gi} is a basis of 𝔤, then there is a dual basisMathworldPlanetmath {gi} with respect to the Killing form, i.e., (gi,gj)=δij. Consider the element Ω=gigi of the universal enveloping algebra of 𝔤. This element, called the Casimir operatorMathworldPlanetmath is centralPlanetmathPlanetmath in the enveloping algebra, and thus commutes with the 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