Editing Universal enveloping algebra (section)

===Example: Pseudo-differential operators===
A key observation during the construction of <math>U(\mathfrak{g})</math> above was that it was a differential algebra, by dint of the fact that any derivation on the Lie algebra can be lifted to <math>U(\mathfrak{g})</math>. Thus, one is led to a ring of [[pseudo-differential operator]]s, from which one can construct Casimir invariants.

If the Lie algebra <math>\mathfrak{g}</math> acts on a space of linear operators, such as in [[Fredholm theory]], then one can construct Casimir invariants on the corresponding space of operators. The quadratic Casimir operator corresponds to an [[elliptic operator]].

If the Lie algebra acts on a differentiable manifold, then each Casimir operator corresponds to a higher-order differential on the cotangent manifold, the second-order differential being the most common and most important.

If the action of the algebra is [[Isometry group|isometric]], as would be the case for [[Riemannian manifold|Riemannian]] or [[pseudo-Riemannian manifold]]s endowed with a metric and the symmetry groups [[SO(N)]] and [[indefinite orthogonal group|SO (P, Q)]], respectively, one can then contract upper and lower indices (with the metric tensor) to obtain more interesting structures. For the quadratic Casimir invariant, this is the [[Laplacian]]. Quartic Casimir operators allow one to square the [[stress–energy tensor]], giving rise to the [[Yang-Mills action]]. The [[Coleman–Mandula theorem]] restricts the form that these can take, when one considers ordinary Lie algebras. However, the [[Lie superalgebra]]s are able to evade the premises of the Coleman–Mandula theorem, and can be used to mix together space and internal symmetries.