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Section 3.7 Casimir operators

Among the linear operators that act on a Lie algebra, there are some, called Casimir operators, that commute with every element of the Lie algebra, and hence reduce to a multiple of the identity transformation. These multiples can depend on the representation under consideration; the resulting constants can therefore be used to label the representations.

An example is the quadratic Casimir operator, which is just (a multiple of) the sum of the squares of an orthonormal basis. For instance, in \(\so(3)\) (see Section 1.2) the standard basis is \(r_m\) and we have

\begin{equation} L^2 = r_x^2 + r_x^2 + r_x^2 = -2 I .\tag{3.7.1} \end{equation}

We can consider \(L^2\) as an operator on \(\su(2)\) (see Section 1.3) using the isomorphism \(\su(2)\cong\so(3)\text{,}\) taking care to map \(r_m\) to \(\frac12 s_m\) in order to preserve commutators. But now we have

\begin{equation} L^2 = \frac14 \left( s_x^2 + s_x^2 + s_x^2 \right) = -\frac34 I .\tag{3.7.2} \end{equation}

The operator \(L^2\) in this case is interpreted as spin, and the eigenvalues take the form \(-\ell(\ell+1)\text{;}\) the defining representation of \(\su(2)\) is a spin-\(\frac12\) representation of \(\su(2)\text{,}\) whereas the defining representation of \(\so(3)\) is a spin-\(1\) representation.

More generally, we can construct the quadratic Casimir operator for any (semisimple) Lie algebra \(\gg\text{.}\) To simplify the notation, write

\begin{equation} g_{mn} = B(e_m,e_n)\tag{3.7.3} \end{equation}

for the components of the Killing form, and let \(g^{pq}\) denote the inverse matrix, so that

\begin{equation} g^{pm} g_{mq} = \delta^p{}_q .\tag{3.7.4} \end{equation}

The quadratic Casimir operator \(\Omega\) is then defined by

\begin{equation} \Omega = g^{mn} e_m e_n\tag{3.7.5} \end{equation}

where the product can be thought of as matrix multiplication, but formally lives in the enveloping algebra of \(\gg\text{.}\) We can now compute

\begin{align} e_m e_n e_q &= e_m e_q e_n + e_m [e_n,e_q]\notag\\ &= e_m e_q e_n + e_m C^p{}_{nq} e_p\notag\\ &= e_q e_m e_n + [e_m,e_q] e_n + e_m C^p{}_{nq} e_p\notag\\ &= e_q e_m e_n + C^p{}_{mq} e_p e_n + e_m C^p{}_{nq} e_p\tag{3.7.6} \end{align}

so that

\begin{align} [\Omega,e_q] &= g^{mn} \left( C^p{}_{mq} e_p e_n + e_m C^p{}_{nq} e_p \right)\notag\\ &= g^{mn} C^p{}_{mq} \left( e_p e_n + e_n e_p \right)\notag\\ &= C_{pmq} \left( e^p e^m + e^m e^p \right) .\tag{3.7.7} \end{align}

In the second equality, we have used the symmetry in the double sum over \(m\) and \(n\) to swap these two indices in the second term, and in the last step we have used the Killing form to raise and lower indices, which preserves symmetry. The resulting term in parentheses is thus symmetric in \(p\) and \(m\text{;}\) we claim (3.6.12) implies that the structure constants (in this index position) are antisymmetric in these two indices, so that (3.7.7) must vanish due to the double sum. Explicitly, when written out in components, (3.6.12) takes the form

\begin{align} 0 &= B\bigl([e_q,e_p],e_m\bigr) + B\bigl(e_p,[e_q,e_m]\bigr)\notag\\ &= C^n{}_{qp} g_{nm} + C^n{}_{qm} g_{pn} = C_{mqp} + C_{pqm} = - \left( C_{mpq} + C_{pmq} \right) .\tag{3.7.8} \end{align}

We conclude that \(\Omega\) commutes with \(e_q\text{,}\) and hence with all elements of \(\gg\text{,}\) as claimed.