= Casimir element
{c}
{wiki}
= Casimir operator
{c}
{synonym}
Let $\mathfrak g$ have a <nondegenerate> symmetric <invariant bilinear form on a Lie algebra> $B$, and choose a <basis> $(x_i)$ with $B$-dual <basis> $(x^i)$. Its quadratic Casimir element is
$$
C_B=\sum_i x_i x^i\in U(\mathfrak g).
$$
The inverse-form <tensor> $\sum_i x_i\otimes x^i$ is independent of the <basis> and invariant under the <Adjoint representation of a Lie algebra>. Multiplying its factors in the <universal enveloping algebra> gives $[z,C_B]=\sum_i([z,x_i]x^i+x_i[z,x^i])=0$, so $C_B$ is central. In a finite-dimensional <irreducible> <complex representation>, the <Schur lemma> makes its represented <quadratic Casimir operator> a scalar.<Lie Groups and Lie Algebras by Pavel Etingof (2024)>[Section 18.3, construction and centrality of the quadratic Casimir]{ref}
With the <Killing form> on a complex <semisimple Lie algebra>, its <Casimir eigenvalue> on <highest weight> $\lambda$ is $(\lambda,\lambda+2\rho)$, with the induced dual <inner product> and <Weyl vector> $\rho$. Scaling $B$ by $c\ne0$ scales $C_B$ by $c^{-1}$. For the <sl2 Lie algebra>, the normalization $B(e,f)=1$, $B(h,h)=2$ gives $ef+fe+h^2/2$; the actual <Killing form> is four times this form, so its Casimir is one quarter of that expression.
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