Casimir element and Casimir operator - Imported Knowlpedia Content

Definition

Let $\mathfrak g$ be a finite-dimensional semisimple Lie algebra over $\mathbb R$ or $\mathbb C$ , let $B$ be its Killing form, and choose bases $(X_i)$ and $(X^i)$ dual with respect to $B$ . The Casimir element in the universal enveloping algebra is

\Omega_B=\sum_i X_iX^i\in U(\mathfrak g).

$\Omega_B$ is independent of the chosen dual bases and belongs to the center of $U(\mathfrak g)$ . If a representation $d\pi$ of $\mathfrak g$ is extended to $U(\mathfrak g)$ , then $d\pi(\Omega_B)$ is its Casimir operator.

Centrality and normalization

The invariance of $B$ makes the tensor corresponding to the identity map on $\mathfrak g$ invariant under the adjoint action; multiplication into $U(\mathfrak g)$ therefore makes $\Omega_B$ central. Replacing $B$ by $cB$ replaces $\Omega_B$ by $c^{-1}\Omega_B$ , so numerical Casimir eigenvalues depend on the normalization of the invariant form Knapp, Chapter V, §5.

Action in representations

Because the Casimir operator commutes with the $\mathfrak g$ -action, it is a module endomorphism. On an irreducible complex representation in a setting where Schur's lemma gives scalar endomorphisms, it acts by a scalar. For highest-weight modules this scalar can be computed from the highest weight and the half-sum of positive roots; the formula changes with the normalization of $B$ .

Conventions and scope

For a real semisimple Lie algebra, one often forms the element after complexifying $\mathfrak g$ . More generally, any nondegenerate invariant symmetric bilinear form produces a quadratic Casimir element by the same construction. “Casimir invariant” may also refer to the scalar eigenvalue or to higher-degree central elements, so it is not always synonymous with this quadratic element.

References

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002. DOI record. Relevant: Chapter V, §5 on the center and Casimir element.
Jacques Dixmier, Enveloping Algebras, Graduate Studies in Mathematics 11, American Mathematical Society, 1996. DOI record. Relevant: Chapters 2 and 7 on enveloping algebras and their centers.