Casimir element

《Lie Groups Beyond an Introduction》 · 印刷第 293 页 Chapter V、Section 4「Complete Reducibility」，式 (5.23) 及 Proposition 5.24 完整陈述；包络代数中的元

For a general semisimple $\mathfrak g$, let $B$ be the Killing form. (To fix the definitions in this section, we shall not allow more general invariant forms in place of $B$.) Let $X_i$ be any basis of $\mathfrak g$ over $\mathbb C$, and let $\widetilde X_i$ be the dual basis relative to $B$, i.e., the basis with $B(\widetilde X_i,X_j)=\delta_{ij}$. The Casimir element $\Omega$ is defined by $\Omega=\sum_{i,j}B(X_i,X_j)\widetilde X_i\widetilde X_j$. Proposition 5.24. In a complex semisimple Lie algebra $\mathfrak g$, the Casimir element $\Omega$ is defined independently of the basis $X_i$ and is a member of the center $Z(\mathfrak g)$ of $U(\mathfrak g)$.

《Introduction to Mathematical Logic, Sixth Edition》 · 印刷第 166 页 Section 3.3.2「Casimir Operator」，式 (3.35) 及命名句；上述构造在表示中的算子语境

Let $\mathfrak g$ be a semisimple Lie algebra. […] Recall that the Killing form $B$ on $\mathfrak g$ is nondegenerate by Theorem 2.5.11. […] Fix a basis $\{X_i\}$ for $\mathfrak g$ and let $\{Y_i\}$ be the $B$-dual basis: $B(X_i,Y_j)=\delta_{ij}$ for all $i,j$. If $(\pi,V)$ is a representation of $\mathfrak g$ (not necessarily finite-dimensional), we define $C_\pi=\sum_i\pi(X_i)\pi(Y_i)$. This linear transformation on $V$ is called the Casimir operator of the representation.