Notes on BRST VI: Casimir Operators

For the case of [tex]G=SU(2)[/tex], it is well-known from the discussion of angular momentum in any quantum mechanics textbook that irreducible representations can be labeled either by j, the highest weight (here, highest eigenvalue of [tex]J_3[/tex] ), or by [tex]j(j+1)[/tex], the eigenvalue of [tex]\mathbf{J\cdot J}[/tex]. The first of these requires making a choice (the z-axis) and looking at a specific vector in the representation, the second doesn’t. It was a physicist (Hendrik Casimir), who first recognized the existence of an analog of [tex]\mathbf{J\cdot J}[/tex] for general semi-simple Lie algebras, and the important role that this plays in representation theory.

The Casimir Operator

Recall that for a semi-simple Lie algebra [tex]\mathfrak g[/tex] one has a non-degenerate, invariant, symmetric bi-linear form [tex](\cdot,\cdot)[/tex], the Killing form, given by

[tex](X,Y)= tr(ad(X)ad(Y))[/tex]

If one starts with [tex]\mathfrak g[/tex] the Lie algebra of a compact group, this bilinear form is defined on [tex]\mathfrak g_{\mathbf C}[/tex], and negative-definite on [tex]\mathfrak g[/tex]. For a simple Lie algebra, taking the trace in a different representation gives the same bilinear form up to a constant. As an example, for the case [tex]\mathfrak g_{\mathbf C}={\mathfrak{sl}(n,\mathbf C)}[/tex], one can show that

([tex]X,Y)=2n\ tr(XY)[/tex]

here taking the trace in the fundamental representation as [tex]n[/tex] by [tex]n[/tex] complex matrices.
One can use the Killing form to define a distinguished quadratic element [tex]\Omega[/tex] of [tex]U(\mathfrak g)[/tex], the Casimir element

[tex]\Omega=\sum_iX_iX^i[/tex]

where [tex]X_i[/tex] is an orthonormal basis with respect to the Killing form and [tex]X^i[/tex] is the dual basis. On any representation [tex]V[/tex], this gives a Casimir operator

[tex]\Omega_V=\sum_i\pi(X_i)\pi(X^i)[/tex]

Note that, taking the representation [tex]V[/tex] to be the space of functions [tex]C^\infty(G)[/tex] on the compact Lie group G, [tex]\Omega_V[/tex] is an invariant second-order differential operator, (minus) the Laplacian.

[tex]\Omega[/tex] is independent of the choice of basis, and belongs to [tex]U(\mathfrak g)^{\mathfrak g}[/tex], the subalgebra of [tex]U(\mathfrak g)[/tex] invariant under the adjoint action. It turns out that [tex]U(\mathfrak g)^{\mathfrak g}=Z(\mathfrak g)[/tex], the center of [tex]U(\mathfrak g)[/tex]. By Schur’s lemma, anything in the center [tex]Z(\mathfrak g)[/tex] must act on an irreducible representation by a scalar. One can compute the scalar for an irreducible representation [tex](\pi,V)[/tex] as follows:

Choose a basis [tex](H_i, X_{\alpha},X_{-\alpha})[/tex] of [tex]\mathfrak g_{\mathbf C}[/tex] with [tex]H_i[/tex] an orthonormal basis of the Cartan subalgebra [tex]\mathfrak t_{\mathbf C}[/tex], and [tex]X_{\pm\alpha}[/tex] elements of [tex]\mathfrak n^{\pm}[/tex] in the [tex]\pm\alpha[/tex] root-spaces of [tex]\mathfrak g_{\mathbf C}[/tex], orthonormal in the sense of satisfying

[tex](X_{\alpha},X_{-\alpha})=1[/tex]

Then one has the following expression for [tex]\Omega[/tex]:

[tex]\Omega=\sum_i H_i^2 + \sum_{+\ roots} (X_{\alpha} X_{-\alpha} +X_{-\alpha}X_{\alpha})[/tex]

To compute the scalar eigenvalue of this on an irreducible representation [tex](\pi,V_{\lambda})[/tex] of highest weight [tex]\lambda[/tex], one can just act on a highest weight vector [tex]v\in V^{\lambda}=V^{\mathfrak n^+}[/tex]. On this vector the raising operators [tex]\pi(X_{\alpha})[/tex] act trivially, and using the commutation relation

[tex][X_{\alpha},X_{-\alpha}]=H_{\alpha}[/tex]

([tex]H_{\alpha}[/tex] is the element of [tex]\mathfrak t_{\mathbf C} [/tex] satisfying [tex](H,H_{\alpha})=\alpha(H)[/tex]) one finds

[tex]\Omega=\sum_i H_i^2 + \sum_{+\ roots}H_{\alpha}= \sum_i H_i^2 +2H_{\rho}[/tex]

where [tex]\rho[/tex] is half the sum of the positive roots, a quantity which keeps appearing in this story. Acting on [tex]v\in V^{\lambda}[/tex] one finds

[tex]\Omega_{V_{\lambda}}v=(\sum_i\lambda(H_i)^2+2\lambda (H_{\rho}))v[/tex]

Using the inner-product [tex]< \cdot,\cdot>[/tex] induced on [tex]\mathfrak t^*[/tex] by the Killing form, this eigenvalue can be written as:

[tex]< \lambda,\lambda>+2< \lambda,\rho>=||\lambda+\rho||^2- ||\rho||^2[/tex]

In the special case [tex]\mathfrak g = \mathfrak {su}(2),\ \mathfrak g_{\mathbf C}=\mathfrak sl(2,\mathbf C)[/tex], there is just one positive root, and one can take

[tex]H_1=h=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\ X_{\alpha}=e=\begin{pmatrix}0&1\\0&0\end{pmatrix},\ X_{-\alpha}=f=\begin{pmatrix}0&0\\1&0\end{pmatrix}[/tex]

Computing the Killing form, one finds

[tex](h,h)=8,\ (e,f)=4[/tex]

and

[tex]\Omega=\frac{1}{8}h^2 + \frac{1}{4}(ef +fe)=\frac{1}{8}h^2 + \frac{1}{4}(h +2fe)[/tex]

On a highest weight vector [tex]\Omega[/tex] acts as

[tex]\Omega=\frac{1}{8}h^2 + \frac{1}{4}h=\frac{1}{8}h(h+2)=\frac{1}{2}(\frac{h}{2}(\frac{h}{2} +1))[/tex]

This is 1/2 times the physicist’s operator [tex]\mathbf{J\cdot J}[/tex], and in the irreducible representation [tex]V_n[/tex] of spin [tex] j=n/2[/tex], it acts with eigenvalue [tex]\frac{1}{2}j(j+1)[/tex].

In the next posting in this series I’ll discuss the Harish-Chandra homomorphism, and the question of how the Casimir acts not just on [tex]V^{\mathfrak n^+}=H^0(\mathfrak n^+,V)[/tex], but on all of the cohomology [tex]H^*(\mathfrak n^+,V)[/tex]. After that, taking note that the Casimir is in some sense a Laplacian, we’ll follow Dirac and introduce Clifford algebras and spinors in order to take its square root.