Casimir invariants for the complete family of quasi-simple orthogonal algebras

A complete choice of generators of the center of the enveloping algebras of real quasi-simple Lie algebras of orthogonal type, for arbitrary dimension, is obtained in a unified setting. The results simultaneously include the well known polynomial invariants of the pseudo-orthogonal algebras so(p, q), as well as the Casimirs for many non-simple algebras such as the inhomogeneous iso(p, q), the Newton–Hooke and Galilei type, etc., which are obtained by contraction(s) starting from the simple algebras so(p, q). The dimension of the center of the enveloping algebra of a quasi-simple orthogonal algebra turns out to be the same as for the simple so(p, q) algebras from which they come by contraction. The structure of the higher order invariants is given in a convenient “pyramidal” manner, in terms of certain sets of “Pauli–Lubanski” elements in the enveloping algebras. As an example showing this approach at work, the scheme is applied to recovering the Casimirs for the (3+1) kinematical algebras. Some prospects on the relevance of these results for the study of expansions are also given.