Casimir Invariant - Definition

Definition

Suppose that g is an n−dimensional semisimple Lie algebra with basis {X₁, ..., X_n}. Moreover let {X1, ..., Xn} be the dual basis of g with respect to a fixed invariant bilinear form (e.g. the Killing form) on g. The Casimir element Ω is an element of the universal enveloping algebra U(g) given by the formula

Although the definition of the Casimir element refers to a particular choice of basis in the Lie algebra, it is easy to show that the resulting element Ω is independent of this choice. Moreover, the invariance of the bilinear form used in the definition implies that the Casimir element commutes with all elements of the Lie algebra g, and hence lies in the center of the universal enveloping algebra U(g).

Given any representation ρ of g on a vector space V, possibly infinite-dimensional, the corresponding Casimir invariant is ρ(Ω), the linear operator on V given by the formula

A special case of this construction plays an important role in differential geometry and global analysis. Suppose that a connected Lie group G with the Lie algebra g acts on a differentiable manifold M, then elements of g are represented by first order differential operators on M. The representation ρ is on the space of smooth functions on M. In this situation the Casimir invariant is the G–invariant second order differential operator on M defined by the above formula.

More general Casimir invariants may also be defined, commonly occurring in the study of pseudo-differential operators in Fredholm theory.