In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the
center of the
universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the representa ...
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 ident ...
. 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 pro ...
, which is a Casimir element of the three-dimensional
rotation group.
More generally, Casimir elements can be used to refer to ''any'' element of the center of the universal enveloping algebra. The
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
of these elements is known to be
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to a
polynomial algebra through the
Harish-Chandra isomorphism
In mathematics, the Harish-Chandra isomorphism, introduced by ,
is an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps the center \mathcal(U(\mathfrak)) of the universal enveloping algebra U(\mathf ...
.
The Casimir element is named after
Hendrik Casimir, who identified them in his description of
rigid body dynamics
In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are '' rigid'' (i.e. they do not deform under the action ...
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 ident ...
. Let ''B'' be a nondegenerate
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
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
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) sho ...
if
is
semisimple
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
.)
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
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the representa ...
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
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
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.
Casimir elements of higher order
The article on
universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the representa ...
s 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
In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
of the
adjoint representation :
:
where is the order of the symmetric tensor
and the
form a
vector space basis
In mathematics, a set of elements of a vector space is called a basis (: bases) if every element of can be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as ...
of
This corresponds to a symmetric homogeneous polynomial
:
in indeterminate variables
in the
polynomial algebra