In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
. A prototypical example is the squared
angular momentum operator
In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum p ...
, which is a Casimir element of the three-dimensional
rotation group.
The Casimir element is named after
Hendrik Casimir
Hendrik Brugt Gerhard Casimir (15 July 1909 – 4 May 2000) was a Dutch physicist best known for his research on the two-fluid model of superconductors (together with C. J. Gorter) in 1934 and the Casimir effect (together with D. Polder) in 1 ...
, 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.
Quadratic Casimir element
Suppose that
is an
-dimensional
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
. Let ''B'' be a nondegenerate
bilinear form on
that is invariant under the
adjoint action of
on itself, meaning that
for all ''X'', ''Y'', ''Z'' in
. (The most typical choice of ''B'' is the
Killing form if
is
semisimple.)
Let
:
be any
basis of
, and
:
be the dual basis of
with respect to ''B''. The Casimir element
for ''B'' is the element of the
universal enveloping algebra given by the formula
:
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
, and hence lies in the
center of the universal enveloping algebra
.
Quadratic Casimir invariant of a linear representation and of a smooth action
Given a
representation ''ρ'' of
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. Suppose that a connected Lie group ''G'' with Lie algebra
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 ''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 In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is giv ...
.
Casimir elements of higher order
The article on
universal enveloping algebras gives a detailed, precise definition of Casimir operators, and an exposition of some of their properties. All Casimir operators correspond to symmetric
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
s in the
symmetric algebra of the
adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is ...
:
:
where is the order of the symmetric tensor
and the
form a
vector space basis of
This corresponds to a symmetric homogeneous polynomial
:
in indeterminate variables
in the
polynomial algebra