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.

examples

applications

homework

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

quiz

See also

• w:Casimir element

references

On paper:

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

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