# Casimir element

Given a simple Lie algebra over the field of complex numbers, the **w:Casimir element** is an element in the universal enveloping algebra 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.

## examples Edit

## applications Edit

## homework Edit

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

## quiz Edit

## See also Edit

## references Edit

*On paper:*

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

*On line: *
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