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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.[1]

Definition

Suppose that \( \mathfrak{g} \) is an n-dimensional semisimple Lie algebra. 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 a fixed invariant bilinear form (e.g. the Killing form) on \(\mathfrak{g} \). The Casimir element \( \Omega \) is an 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 of the Casimir element refers to a particular choice of basis in the Lie algebra, it is easy to show that the resulting element Ω is independent of this choice. Moreover, the invariance of the bilinear form used in the definition 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 \(U(\mathfrak{g}) \).

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

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

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

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

Properties

The Casimir operator is a distinguished 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.

The number of independent elements of the center of the universal enveloping algebra is also the rank in the case of a semisimple Lie algebra. 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

References

Oliver, David (2004). The shaggy steed of physics: mathematical beauty in the physical world. Springer. p. 81. ISBN 978-0-387-40307-6.