Casimir element

In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the centre of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group.

The Casimir element is named after Hendrik Casimir, who identified them in his description of rigid body dynamics in 1931.

Definition

Suppose that $\mathfrak{g}$ is an $n$-dimensional semisimple Lie algebra. Let B be a bilinear form on $\mathfrak{g}$ that is invariant under the adjoint action of $\mathfrak{g}$ on itself, meaning that $B(ad_XY,Z)+B(Y,ad_XZ)=0$ for all X,Y,Z in G. (The most typical choice of B is the Killing form.) Let

$\{X_i\}_{i=1}^n$

be any basis of $\mathfrak{g}$, and

$\{X^i\}_{i=1}^n$

be the dual basis of $\mathfrak{g}$ with respect to B. The Casimir element $\Omega$ for B is the element of the universal enveloping algebra $U(\mathfrak{g})$ given by the formula

$\Omega = \sum_{i=1}^n X_i X^i.$

Although the definition relies on a choice of basis for the Lie algebra, it is easy to show that Ω is independent of this choice. On the other hand, Ω does depend on the bilinear form B. The invariance of B implies that the Casimir element commutes with all elements of the Lie algebra $\mathfrak{g}$, and hence lies in the center of the universal enveloping algebra $U(\mathfrak{g})$.

Casimir invariant of a linear representation and of a smooth action

Given a representation ρ of $\mathfrak{g}$ on a vector space V, possibly infinite-dimensional, the Casimir invariant of ρ is defined to be ρ(Ω), the linear operator on V given by the formula

$\rho(\Omega) = \sum_{i=1}^n \rho(X_i)\rho(X^i).$

Here we are assuming that B is the Killing form, otherwise B must be specified.

A specific form of this construction plays an important role in differential geometry and global analysis. Suppose that a connected Lie group G with Lie algebra $\mathfrak{g}$ acts on a differentiable manifold M. Consider the corresponding representation ρ of G on the space of smooth functions on M. Then elements of $\mathfrak{g}$ are represented by first order differential operators on M. In this situation, the Casimir invariant of ρ is the G-invariant second order differential operator on M defined by the above formula.

Specializing further, if it happens that M has a Riemannian metric on which G acts transitively by isometries, and the stabilizer subgroup G_x of a point acts irreducibly on the tangent space of M at x, then the Casimir invariant of ρ is a scalar multiple of the Laplacian operator coming from the metric.

More general Casimir invariants may also be defined, commonly occurring in the study of pseudo-differential operators in Fredholm theory.

Properties

Uniqueness

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

Generalizations

The Casimir operator is a distinguished quadratic element of the center of the universal enveloping algebra of the Lie algebra. In other words, it is a member of the algebra of all differential operators that commutes with all the generators in the Lie algebra. In fact all quadratic elements in the center of the universal enveloping algebra arise this way. However, the center may contain other, non-quadratic, elements.

By Racah's theorem, for a semisimple Lie algebra the dimension of the center of the universal enveloping algebra is equal to its rank. The Casimir operator gives the concept of the Laplacian on a general semisimple Lie group; but this way of counting shows that there may be 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, the Casimir operator is thus proportional to the identity. This constant of proportionality 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 constants, 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..

Example: so(3)

The Lie algebra $\mathfrak{so}(3)$ 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 $L_x,\, L_y,\, L_z$ of the algebra. That is, the Casimir invariant is given by

$L^2=L_x^2+L_y^2+L_z^2.$

In an irreducible representation, the invariance of the Casimir operator implies that it is a multiple of the identity element e of the algebra, so that

$L^2=L_x^2+L_y^2+L_z^2=\ell(\ell+1)e.$

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

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

$L_x= \begin{pmatrix} 0& 0& 0\\ 0& 0& -1\\ 0& 1& 0 \end{pmatrix}, L_y= \begin{pmatrix} 0& 0& 1\\ 0& 0& 0\\ -1& 0& 0 \end{pmatrix}, L_z= \begin{pmatrix} 0& -1& 0\\ 1& 0& 0\\ 0& 0& 0 \end{pmatrix}.$

The quadratic Casimir invariant is then

$L^2=L_x^2+L_y^2+L_z^2= 2 \begin{pmatrix} 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1 \end{pmatrix}$

as $\ell(\ell+1)\,=\,2$ when $\ell\,=\,1$. Similarly, the two dimensional representation has a basis given by the Pauli matrices, which correspond to spin 1/2.

Eigenvalues

Given that $\Omega$ is central in the enveloping algebra, it acts on simple modules by a scalar. Let $\langle,\rangle$ be any bilinear symmetric non-degenerate form, by which we define $\Omega$. Let $L(\lambda)$ be the finite dimensional highest weight module of weight $\lambda$. Then the Casimir element $\Omega$ acts on $L(\lambda)$ by the constant $\langle \lambda, \lambda + 2 \rho \rangle,$ where $\rho$ is the weight defined by half the sum of the positive roots.

See also

• Harish-Chandra isomorphism
• Pauli–Lubanski pseudovector