Departamento de Geometria y Topologia,

Faculted de Ciencias Matematicas,

Universidad Complutense de Madrid,

Plaza Ciencias 3,

E-28040 Madrid,

SPAIN

E-mail: rutwig@mat.ucm.es, otto_campoamor@mat.ucm.es

**Extension of Gel'fand matrix method to the Casimir
operators of inhomogeneous groups**

**Abstract:**

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.