# 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 .

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 .

How to Cite This Entry:
Casimir element. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Casimir_element&oldid=11861
This article was adapted from an original article by D.P. Zhelobenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article