# Casimir element

Given a simple Lie algebra ${\displaystyle {\mathfrak {g}}}$over the field ${\displaystyle \mathbb {C} }$ of complex numbers, the w:Casimir element is an element in the universal enveloping algebra ${\displaystyle U({\mathfrak {g}})}$ that commutes with every other element there. In fact it generates the whole centre of the algebra. It is constructed from the invariant bilinear form (w:Cartan-Killing form) and is pivotal in the structure theory of representations of Lie algebras.

## homework

1. write down the casimir element for sl2 and show that it commutes with the standard generators e,f,h.

## references

On paper:

• James Humphreys: Introduction to Lie algebras and representation theory, ISBN 9780387900537 , pp.27,118

On line: ...