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I read about the Casimir element just recently and I found it a bit difficult to wrap my mind around the definition(s). In fact, I have seen two different definitions. For concreteness, let $\mathfrak{g}$ be the Lie algebra of a connected (semi-simple) Lie group $G$, and $\pi\colon G\to GL(V)$ a group representation of $G$ on an $n$-dimensional vector space $V$. (I assume for simplicity that an orthonormal basis relative to the Killing form exists.)

  1. (Definition as a matrix using the derived representation) For an orthonormal basis $\{X_i\}$ of $\mathfrak{g}$, the Casimir element is defined to be the matrix $\Omega=\sum d\pi(X_i)^2$ in $\text{Mat}(n\times n)$.

  2. (Definition as an element in the center of enveloping Lie algebra) For an orthonormal basis $\{X_i\}$ of $\mathfrak{g}$ the Casimir element is defined by $\Omega=\sum X_i^2$ in $\mathfrak{U}(\mathfrak{g})$.

My question is how are these two definitions related? For instance, if $\pi$ is irreducible, it is obvious to me from Schur's Lemma that $\Omega$ in the first definition is a scalar matrix. Can this result be somehow transferred to the $\Omega$ in the second definition? More generally, is any of these closer to `the right way' of thinking about the Casimir element?

Any explanation is appreciated!

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The second way is closer to rigth understanding of the Casimir elements. This approаch can be extended to define a generalized Casimir elements: Let $\{u_i \}$ and $\{u_i^* \}$ be basis of two dual representations of Lie algebra $g$ in its universal enveloping agebra $\mathfrak{g}$. Then the element $$ \sum_i u_i u^*_i, $$ lies in the center $Z(g)$ and called the generalised Casimir element.

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  • $\begingroup$ Thanks for your answer. In the last line I guess you mean $Z(\mathfrak{U}(\mathfrak{g})$. Do you have any thoughts on the relation between the two definitions that I have written in my question? $\endgroup$
    – EPS
    Oct 6, 2014 at 0:37

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