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
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
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 representati ...
of a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
. 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 prob ...
, which is a Casimir element of the three-dimensional
rotation group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
.
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 19 ...
, 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 of a ...
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 Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
. 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 linear i ...
on
that is invariant under the
adjoint action
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 GL( ...
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) show ...
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
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
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 representati ...
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
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
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
The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
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 or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
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
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
coming from the metric.
More general Casimir invariants may also be defined, commonly occurring in the study of
pseudo-differential operator
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in ...
s 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 given ...
.
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 representati ...
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; t ...
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
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 GL(n ...
:
:
where is the order of the symmetric tensor
and the
form a
vector space basis
In mathematics, a Set (mathematics), set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referre ...
of
This corresponds to a symmetric homogeneous polynomial
:
in indeterminate variables
in the
polynomial algebra
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...