In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center 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.
Contents
- Definition
- Quadratic Casimir element
- Casimir invariant of a linear representation and of a smooth action
- General case
- Uniqueness
- Relation to the Laplacian on G
- Generalizations
- Example s o 3 displaystyle mathfrak so3
- Eigenvalues
- References
The Casimir element is named after Hendrik Casimir, who identified them in his description of rigid body dynamics in 1931.
Definition
The most commonly-used Casimir invariant is the quadratic invariant. It is the simplest to define, and so is given first. However, one may also have Casimir invariants of higher order, which correspond to homogeneous symmetric polynomials of higher order; their definition is given last.
Quadratic Casimir element
Suppose that
be any basis of
be the dual basis of
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
Casimir invariant of a linear representation and of a smooth action
Given a representation ρ of
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
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.
General case
The article on universal enveloping algebras gives a detailed, precise definition of Casimir operators, and an exposition of some of their properties. In particular, all Casimir operators correspond to symmetric homogeneous polynomials in the symmetric algebra of the adjoint representation
where m is the order of the symmetric tensor
in m indeterminate variables
Not just any symmetric tensor (symmetric homogeneous polynomial) will do; it must explicitly commute with the Lie bracket! That is, one must have that
for all basis elements
in order to obtain
This result is originally due to Israel Gelfand. The commutation relation implies that the Casimir operators lie in the center of the universal enveloping algebra, and, in particular, always commute with any element of the Lie algebra. It is due to this property of commutation that allows a representation of a Lie algebra to be labelled by eigenvalues of the associated Casimir operators.
Note also that any linear combination of the symmetric polynomials described above will lie in the center as well: therefore, the Casimir operators are, by definition, restricted to that subset that span this space (that provide a basis for this space). For a semisimple Lie algebra of rank r, there will be r Casimir invariants.
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
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: s o ( 3 ) {displaystyle {mathfrak {so}}(3)}
The Lie algebra
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
In quantum mechanics, the scalar value
For a given value of
The quadratic Casimir invariant is then
as
Eigenvalues
Given that