Casimir goes to Casimir

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In mathematics, "Casimir goes to Casimir" is an aphorism expressing the centrality and universality of the Casimir operator. This can be summarized as follows:

Given any representation $V$ of a semisimple Lie algbera $\mathfrak{g}$ , the Casimir operator induces a $\mathfrak{g}$ -module homomorphism $Ω$ of $V$ (as follows from the centrality of the Casimir operator). Let now $V 1, V 2$ be two representations of $\mathfrak{g}$ with corresponding Casimir operators $Ω 1,Ω 2$ , and $f:V_1\to V_2$ a $\mathfrak{g}$ -module homomorphism. Then since $Ω$ is a member of the universal enveloping algebra, it commutes with $f$ , i.e. $f\circ \Omega_1=\Omega_2\circ f$ . Stated less rigorously, Casimir goes to Casimir under any $\mathfrak{g}$ -module homomorphism.