Casimir element

Properties

Uniqueness of the quadratic Casimir element

Since for a simple Lie algebra every invariant bilinear form is a multiple of the Killing form, the corresponding Casimir element is uniquely defined up to a constant. For a general semisimple Lie algebra, the space of invariant bilinear forms has one basis vector for each simple component, and hence the same is true for the space of corresponding Casimir operators.

Relation to the Laplacian on G

If ${\displaystyle G}$ is a compact Lie group with Lie algebra ${\displaystyle \mathfrak{g}}$, the choice of a nondegenerate invariant bilinear form on ${\displaystyle \mathfrak{g}}$ corresponds to a choice of bi-invariant Riemannian metric on ${\displaystyle G}$. Then under the identification of the universal enveloping algebra of ${\displaystyle \mathfrak{g}}$ with the left invariant differential operators on ${\displaystyle G}$, the Casimir element of the bilinear form on ${\displaystyle \mathfrak{g}}$ maps to the Laplacian of ${\displaystyle G}$ (with respect to the corresponding bi-invariant metric).

Casimir elements and representation theory

By Racah's theorem, for a semisimple Lie algebra the dimension of the center of its universal enveloping algebra is equal to its rank. The Casimir operator gives the concept of the Laplacian on a general semisimple Lie group; but there is no unique analogue of the Laplacian, for rank > 1.

By definition any member of the center of the universal enveloping algebra commutes with all other elements in the algebra. By Schur's Lemma, in any irreducible representation of the Lie algebra, any Casimir element is thus proportional to the identity. The eigenvalues of all Casimir elements can be used to classify the representations of the Lie algebra (and hence, also of its Lie group).

Physical mass and spin are examples of these eigenvalues, as are many other quantum numbers found in quantum mechanics. Superficially, topological quantum numbers form an exception to this pattern; although deeper theories hint that these are two facets of the same phenomenon..

Let ${\displaystyle L(\lambda )}$ be the finite dimensional highest weight module of weight ${\displaystyle \lambda }$. Then the quadratic Casimir element ${\displaystyle \mathrm{\Omega }}$ acts on ${\displaystyle L(\lambda )}$ by the constant

${\displaystyle ⟨\lambda ,\lambda +2\rho ⟩=⟨\lambda +\rho ,\lambda +\rho ⟩-⟨\rho ,\rho ⟩,}$

where ${\displaystyle \rho }$ is the weight defined by half the sum of the positive roots. If ${\displaystyle L(\lambda )}$ is nontrivial (i.e. if ${\displaystyle \lambda \ne 0}$), then this constant is nonzero. After all, since ${\displaystyle \lambda }$ is dominant, if ${\displaystyle \lambda \ne 0}$, then ${\displaystyle ⟨\lambda ,\lambda ⟩>0}$ and ${\displaystyle ⟨\lambda ,\rho ⟩\ge 0}$, showing that ${\displaystyle ⟨\lambda ,\lambda +2\rho ⟩>0}$. This observation plays an important role in the proof of Weyl's theorem on complete reducibility. It is also possible to prove the nonvanishing of the eigenvalue in a more abstract way—without using an explicit formula for the eigenvalue—using Cartan's criterion; see Sections 4.3 and 6.2 in the book of Humphreys.

Symmetric invariant tensors of simple Lie algebras

A Casimir element of order ${\displaystyle m}$ corresponds to a symmetric invariant tensor of the same order via ${\displaystyle C_{(m)}={\kappa }^{i_1{i}_2\cdots i_m}{X}_{i_1}{X}_{i_2}\cdots X_{i_m}}$. Constructing and relating Casimir elements is equivalent to doing the same for symmetric invariant tensors.

Construction of symmetric invariant tensors

Symmetric invariant tensors may be constructed as symmetrized traces in the defining representation

${\displaystyle k_{i_1{i}_2\cdots i_m}^{(m)}=\text{Tr}\left(X_{(i_1}{X}_{i_2}\cdots X_{i_m)}\right)}$

where indices are raised and lowered by the Killing form, and symmetrized under all permutations.

It is also possible to construct symmetric invariant tensors from the antisymmetric invariant tensors of the type

${\displaystyle {\mathrm{\Omega }}_{i_1{i}_2\cdots i_{2m-1}}^{(2m-1)}=f_{i_1[i_2}^{j_1}\cdots f_{i_{2m-3}{i}_{2m-2}]}^{j_{m-1}}{k}_{j_1\cdots j_{m-1}{i}_{2m-1}}^{(m)}}$

The symmetric invariant tensor

${\displaystyle t_{i_1{i}_2\cdots i_m}^{(m)}={\mathrm{\Omega }}_{j_1{j}_2\cdots j_{2m-2}{i}_m}^{(2m-1)}{f}_{i_1}^{j_1{j}_2}\cdots f_{i_{m-1}}^{j_{2m-2}{j}_{2m-3}}}$

is traceless for ${\displaystyle m>2}$. Such invariant tensors are orthogonal to one another in the sense that ${\displaystyle t_{i_1{i}_2\cdots i_m}^{(m)}{\left(t^{(n)}\right)}^{i_1{i}_2\cdots i_m{i}_{m+1}\cdots i_n}=0}$ if ${\displaystyle n>m}$.

In the case of the simple Lie algebra ${\displaystyle A_l={\mathfrak{s}\mathfrak{l}}_{l+1}}$, let us introduce the fully symmetric tensor of order three ${\displaystyle d_{ijk}}$ such that, in the defining representation,

${\displaystyle X_i{X}_j=\frac{2}{\ell +1}{\delta }_{ij}+f_{ij}^k{X}_k+d_{ij}^k{X}_k}$

Then the Sudbery symmetric invariant tensors are

${\displaystyle d_{i_1{i}_2}^{(2)}={\delta }_{i_1{i}_2}}$

${\displaystyle d_{i_1{i}_2{i}_3}^{(3)}=d_{i_1{i}_2{i}_3}}$

${\displaystyle d_{i_1{i}_2{i}_3{i}_4}^{(4)}=d_{(i_1{i}_2}{}^j{d}_{i_3{i}_4)j}}$

${\displaystyle d_{i_1{i}_2{i}_3{i}_4{i}_5}^{(5)}=d_{(i_1{i}_2}{}^j{d}^j{}_{i_3}{}^k{d}_{i_4{i}_5)k}}$

Relations between symmetric invariant tensors

For a simple Lie algebra of rank ${\displaystyle r}$, there are ${\displaystyle r}$ algebraically independent symmetric invariant tensors. Therefore, any such tensor can be expressed in terms of ${\displaystyle r}$ given tensors. There is a systematic method for deriving complete sets of identities between symmetric invariant tensors.

In the case of the Lie algebra ${\displaystyle A_l}$, the symmetric invariant tensors ${\displaystyle t^{(m)}}$ obey ${\displaystyle t^{(m>l+1)}=0}$. Reexpressing these tensors in terms of other families such as ${\displaystyle d^{(m)}}$ or ${\displaystyle k^{(m)}}$ gives rise to nontrivial relations within these other families. For example, the Sudbery tensors ${\displaystyle d^{(m>l+1)}}$ may be expressed in terms of ${\displaystyle d^{(2)},\cdots ,d^{(l+1)}}$, with relations of the type

${\displaystyle d_{i_1{i}_2{i}_3{i}_4}^{(4)}\underset{l=2}{=}\frac{1}{3}{\delta }_{(i_1{i}_2}{\delta }_{i_3{i}_4)}}$

${\displaystyle d_{i_1{i}_2{i}_3{i}_4{i}_5}^{(5)}\underset{l=2}{=}\frac{1}{3}{d}_{(i_1{i}_2{i}_3}{\delta }_{i_4{i}_5)}}$

${\displaystyle d_{i_1{i}_2{i}_3{i}_4{i}_5}^{(5)}\underset{l=3}{=}\frac{2}{3}{d}_{(i_1{i}_2{i}_3}{\delta }_{i_4{i}_5)}}$

Structure constants also obey identities that are not directly related to symmetric invariant tensors, for example

${\displaystyle 3d_{ab}{}^e{d}_{cde}-f_{ac}{}^e{f}_{bde}-f_{ad}{}^e{f}_{bce}\underset{l=2}{=}{\delta }_{ac}{\delta }_{bd}+{\delta }_{ad}{\delta }_{bc}-{\delta }_{ab}{\delta }_{cd}}$

Examples

Case of sl(2)

The Lie algebra ${\displaystyle {\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})}$ consists of two-by-two complex matrices with zero trace. There are three standard basis elements, ${\displaystyle e}$,${\displaystyle f}$, and ${\displaystyle h}$, with

${\displaystyle \begin{array}{cccccc}e& =\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right],& f& =\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right],& h& =\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right].\end{array}}$

The commutators are

${\displaystyle \begin{array}{cccccc}[e,f]& =h,& [h,f]& =-2f,& [h,e]& =2e.\end{array}}$

One can show that the Casimir element is

$${\displaystyle \mathrm{\Omega }=ef+fe+\frac{1}{2}{h}^2=\frac{1}{2}{h}^2+h+2fe.}$$

This expression lives in the universal enveloping algebra ${\displaystyle U({\mathfrak{s}\mathfrak{l}}_2(\mathbb{C}))}$, not the algebra of ${\displaystyle 2\times 2}$ matrices, so care should be taken not to plug the above ${\displaystyle 2\times 2}$ matrices into this formula and use matrix multiplication to simplify.

As an example of how the Casimir element acts on a representation, consider the adjoint representation of ${\displaystyle {\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})}$ on itself. Using the ordered basis ${\displaystyle (e,h,f)}$ for ${\displaystyle {\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})}$, we can express the adjoint action of any element as a ${\displaystyle 3\times 3}$ matrix. In particular:

${\displaystyle \begin{array}{cccccc}{\mathsf{A}\mathsf{d}}_e& =\left[\begin{array}{ccc}0& -2& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right],& {\mathsf{A}\mathsf{d}}_f& =\left[\begin{array}{ccc}0& 0& 0\\ -1& 0& 0\\ 0& 2& 0\end{array}\right],& {\mathsf{A}\mathsf{d}}_h& =\left[\begin{array}{ccc}2& 0& 0\\ 0& 0& 0\\ 0& 0& -2\end{array}\right]\end{array}}$

The Casimir element then acts on this representation by the matrix

${\displaystyle \begin{array}{cc}{\mathsf{A}\mathsf{d}}_{\mathrm{\Omega }}& ={\mathsf{A}\mathsf{d}}_e{\mathsf{A}\mathsf{d}}_f+{\mathsf{A}\mathsf{d}}_f{\mathsf{A}\mathsf{d}}_e+\frac{1}{2}{\mathsf{A}\mathsf{d}}_h^2\\ & =\left[\begin{array}{ccc}0& -2& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]\left[\begin{array}{ccc}0& 0& 0\\ -1& 0& 0\\ 0& 2& 0\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ -1& 0& 0\\ 0& 2& 0\end{array}\right]\left[\begin{array}{ccc}0& -2& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]+\frac{1}{2}{\left[\begin{array}{ccc}2& 0& 0\\ 0& 0& 0\\ 0& 0& -2\end{array}\right]}^2\\ & =\left[\begin{array}{ccc}4& 0& 0\\ 0& 4& 0\\ 0& 0& 4\end{array}\right]\end{array}}$

Case of so(3)

The Lie algebra ${\displaystyle \mathfrak{s}\mathfrak{o}(3)}$ is the Lie algebra of SO(3), the rotation group for three-dimensional Euclidean space. It is simple of rank 1, and so it has a single independent Casimir. The Killing form for the rotation group is just the Kronecker delta, and so the Casimir invariant is simply the sum of the squares of the generators ${\displaystyle L_x,L_y,L_z}$ of the algebra. That is, the Casimir invariant is given by

${\displaystyle L^2=L_x^2+L_y^2+L_z^2.}$

Consider the irreducible representation of ${\displaystyle \mathfrak{s}\mathfrak{o}(3)}$ in which the largest eigenvalue of ${\displaystyle L_z}$ is ${\displaystyle \ell }$, where the possible values of ${\displaystyle \ell }$ are $0,\frac{1}{2},1,\frac{3}{2},\dots$. The invariance of the Casimir operator implies that it is a multiple of the identity operator ${\displaystyle I}$. This constant can be computed explicitly, giving the following result

${\displaystyle L^2=L_x^2+L_y^2+L_z^2=\ell (\ell +1)I.}$

In quantum mechanics, the scalar value ${\displaystyle \ell }$ is referred to as the total angular momentum. For finite-dimensional matrix-valued representations of the rotation group, ${\displaystyle \ell }$ always takes on integer values (for bosonic representations) or half-integer values (for fermionic representations).

For a given value of ${\displaystyle \ell }$, the matrix representation is ${\displaystyle (2\ell +1)}$-dimensional. Thus, for example, the three-dimensional representation for ${\displaystyle \mathfrak{s}\mathfrak{o}(3)}$ corresponds to ${\displaystyle \ell =1}$, and is given by the generators

${\displaystyle \begin{array}{cccccc}{L}_x& =i\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& -1\\ 0& 1& 0\end{array}\right);& L_y& =i\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ -1& 0& 0\end{array}\right);& L_z& =i\left(\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right),\end{array}}$

where the factors of ${\displaystyle i}$ are needed for agreement with the physics convention (used here) that the generators should be skew-self-adjoint operators.

The quadratic Casimir invariant can then easily be computed by hand, with the result that

${\displaystyle L^2=L_x^2+L_y^2+L_z^2=2\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)}$

as ${\displaystyle \ell (\ell +1)=2}$ when ${\displaystyle \ell =1}$.

This is what is meant when we say that the eigenvalues of the Casimir operator are used to classify the irreducible representations of a Lie algebra (and of an associated Lie group): two irreducible representations of a Lie Algebra are equivalent if and only if their Casimir element have the same eigenvalue. In this case, the irreps of ${\displaystyle \mathfrak{s}\mathfrak{o}(3)}$ are completely determined by the value of ${\displaystyle \ell }$, or equivalently, by the value of ${\displaystyle \ell (\ell +1)}$. Similarly, the two dimensional representation has a basis given by the Pauli matrices, which correspond to spin 1⁄2, and one can again check the formula for the Casimir by direct computation.