A prototypical example is the squared angular momentum operator, which is a Casimir element of the three-dimensional rotation group.
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 A specific form of this construction plays an important role in differential geometry and global analysis.
acts on a differentiable manifold M. Consider the corresponding representation ρ of G on the space of smooth functions on M. Then elements of
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 Gx 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.
The article on universal enveloping algebras gives a detailed, precise definition of Casimir operators, and an exposition of some of their properties.
This corresponds to a symmetric homogeneous polynomial in m indeterminate variables
over a field K. The reason for the symmetry follows from the PBW theorem and is discussed in much greater detail in the article on universal enveloping algebras.
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.
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 Schur's Lemma, in any irreducible representation of the Lie algebra, any Casimir element is thus proportional to the identity.
Superficially, topological quantum numbers form an exception to this pattern; although deeper theories hint that these are two facets of the same phenomenon.
is the weight defined by half the sum of the positive roots.
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.
Constructing and relating Casimir elements is equivalent to doing the same for symmetric invariant tensors.
such that, in the defining representation, Then the Sudbery symmetric invariant tensors are[6] For a simple Lie algebra of rank
There is a systematic method for deriving complete sets of identities between symmetric invariant tensors.
gives rise to nontrivial relations within these other families.
, with relations of the type[7] Structure constants also obey identities that are not directly related to symmetric invariant tensors, for example[8] The Lie algebra
consists of two-by-two complex matrices with zero trace.
is the Lie algebra of SO(3), the rotation group for three-dimensional Euclidean space.
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
This constant can be computed explicitly, giving the following result[9] In quantum mechanics, the scalar value
are needed for agreement with the physics convention (used here) that the generators should be skew-self-adjoint operators.
[10] The quadratic Casimir invariant can then easily be computed by hand, with the result that as
This is what is meant when we say that the eigenvalues of the Casimir operator is 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.
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.