Casimir element

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Casimir operator

A central element of special form in the universal enveloping algebra of a semi-simple Lie algebra. Such operators were first introduced, for a particular case, by H. Casimir [1].

Let be a semi-simple finite-dimensional Lie algebra over a field of characteristic , and let be an invariant symmetric bilinear form on (that is, for all ) which is non-degenerate on a Cartan subalgebra . Then a Casimir element of the Lie algebra with respect to the form is an element of the universal enveloping algebra that is representable in the form

Here , are dual bases of with respect to , that is, , , where is the Kronecker symbol and . The element does not depend on the choice of the dual bases in and belongs to the centre of . If is a simple algebra, then a Casimir element of defined by the Killing form is the unique (up to a scalar multiplier) central element in that is representable as a homogeneous quadratic polynomial in the elements of .

Every linear representation of a semi-simple algebra in a finite-dimensional space defines an invariant symmetric bilinear form

on , which is non-degenerate on the subalgebra complementary to , and therefore also defines some Casimir element . If is an irreducible representation, then the extension of onto takes into .


[1] H. Casimir, B.L. van der Waerden, "Algebraischer Beweis der Vollständigen Reduzibilität der Darstellungen halbeinfacher Liescher Gruppen" Math. Ann. , 111 (1935) pp. 1–2
[2] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)
[3] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)
[4] N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979))
[5] M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian)
[6] J. Dixmier, "Enveloping algebras" , North-Holland (1977) (Translated from French)


The Casimir element determined by is called the Casimir element of the linear representation .

An additional good reference is [a1].


[a1] J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972)
How to Cite This Entry:
Casimir element. D.P. Zhelobenko (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098