I am a bit confused with the information that is provided by the Casimir operator. First, with my understanding, a Casimir operator is defined as, $$\Omega_\rho := \sum_{i-1}^{dim L} \rho(X_i) \circ \rho(X^i),$$ where $\rho: L \to End(V)$ is a (faithful) representation of a Lie algebra $L$ with basis ${X_i}$ and $V$ is a vector space. The ${X^i}$ form the dual basis of $L^\star$ with respect to the Killing form. Now, it seems to be a well known fact that for irreps, the Casimir operator is just $\Omega_\rho = c_\rho id_V$ and that in fact $c_\rho = dim L/dim V$. At first glance, it is then not clear to me what the Casimir operator actually provides from a physical point of view. Matters are made worse when I see in the literature that there sometimes seems to be multiple such operators. A notable example is that in the BHTZ paper, we have $\textit{two}$ Casimir invariants, $$I_1 = \omega^{ab} \omega_{ab}, \qquad I_2 = \epsilon^{abcd} \omega_{ab} \omega_{cd},$$ where the $\omega_{ab}$ are elements of $so(2,2)$. The first definition seems to at least resemble the definition that I give for the Casimir operator, although it is different as this uses generic elements in the algebra instead of basis elements; and where does the second one really come from? Finally, the $I_1$ and $I_2$ are also directly related to the mass $M$ and angular momentum $J$ of the BTZ black hole respectively, i.e. the gauge invariant parts of the black hole which is not only unclear for me how this can be deduced from the definition of the Casimir but also that $M, J$ can really take any value and not some prescribed value as stated above. What is going on?


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