Extension of Gel'fand matrix method to the Casimir operators of inhomogeneous groups
In 1950 I.M. Gel'fand showed how to use the generic matrix of standard representation of orthogonal groups to determine the Casimir operators of the algebra by means of characteristic polynomials. Similar methods have been developed for other semisimple Lie algebras by various authors. From the construction it seems necessary that the used matrix corresponds to some faithful representation of the algebra. We show that the essence of the Gel'fand method can be extended to other non-semisimple Lie algebras, specially the class of inhomogeneous Lie algebras, using extended matrices that also correspond to the standard representation. Considering linear combinations of the characteristic polynomials of the representation matrices and its minors, the Casimir operators of the inhomogeneous algebras can be recovered. It is moreover shown that even in the case where the matrices correspond to non-faithful representations (as happens when In\"on\"u-Wigner contractions are considered), the method remains effective. The extreme case where the employed matrix has no interpretation in terms of representations of Lie algebras is also proved to exist. The method moreover provides an insight to the problem why the contraction of independent Casimir invariants are not necessarily independent invariants of the contractions, and how to derive an alternative procedure based on PDEs to reconstruct additional independent invariants of the contractions. Applications to the invariant operators of subalgebra chains are given, and a sufficiency criterion to obtain certain solutions to the missing label problem is developed.