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 \mathfrak{g} is an n-dimensional
Lie algebra. Let
B be a nondegenerate
bilinear form on \mathfrak{g} that is invariant under the
adjoint action of \mathfrak{g} on itself, meaning that B(\operatorname{ad}_XY, Z) + B(Y, \operatorname{ad}_X Z) = 0 for all
X,
Y,
Z in \mathfrak{g}. (The most typical choice of
B is the
Killing form if \mathfrak{g} is
semisimple.) Let :\{X_i\}_{i=1}^n be any
basis of \mathfrak{g}, and :\{X^i\}_{i=1}^n be the dual basis of \mathfrak{g} with respect to
B. The
Casimir element \Omega for
B is the element of the
universal enveloping algebra U(\mathfrak{g}) given by the formula :\Omega = \sum_{i=1}^n X_i X^i. 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 \mathfrak{g}, and hence lies in the
center of the universal enveloping algebra U(\mathfrak{g}).
Quadratic Casimir invariant of a linear representation and of a smooth action Given a
representation ρ of \mathfrak{g} 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 :\rho(\Omega) = \sum_{i=1}^n \rho(X_i)\rho(X^i). 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 \mathfrak{g}
acts on a
differentiable manifold M. Consider the corresponding representation ρ of
G on the space of smooth functions on M. Then elements of \mathfrak{g} 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.
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 polynomials in the
symmetric algebra of the
adjoint representation \operatorname{ad}_\mathfrak{g}.: :C_{(m)} = \kappa^{ij\cdots k} X_i \otimes X_j \otimes \cdots\otimes X_k where is the order of the symmetric tensor \kappa^{ij\cdots k} and the X_i form a
vector space basis of \mathfrak{g}. This corresponds to a symmetric homogeneous polynomial :c_{(m)} = \kappa^{ij\cdots k} t_i t_j \cdots t_k in indeterminate variables t_i in the
polynomial algebra K[t_i, t_j, \cdots ,t_k] over a field The reason for the symmetry follows from the
PBW theorem and is discussed in much greater detail in the article on
universal enveloping algebras. Moreover, a Casimir element must belong to the center of the universal enveloping algebra, i.e. it must obey :[C_{(m)}, X_i] = 0 for all basis elements X_i. In terms of the corresponding symmetric tensor \kappa^{ij\cdots k}, this condition is equivalent to the tensor being invariant: :f_{ij}^{\;\; k} \kappa^{jl\cdots m} + f_{ij}^{\;\; l} \kappa^{kj\cdots m} + \cdots + f_{ij}^{\;\; m} \kappa^{kl\cdots j} = 0 where f_{ij}^{\;\; k} are the
structure constants of the Lie algebra i.e. [X_i,X_j]=f_{ij}^{\;\; k}X_k. == Properties ==