Determinantal formulae for the Casimir operators of inhomogeneous Lie algebras

Abstract

Contractions of Lie algebras are combined with the classical matrix method of Gel'fand to obtain matrix formulae for the Casimir operators of inhomogeneous Lie algebras. The method is presented for the inhomogeneous pseudo-unitary Lie algebras $I\frak{u}(p,q)$. This procedure is extended to contractions of $I\frak{u}(p,q)$ isomorphic to an extension by a derivation of the inhomogeneous special pseudo-unitary Lie algebras $I\frak{su}(p-1,q)$, providing an additional analytical method to obtain their invariants. Further, matrix formulae for the invariants of other inhomogeneous Lie algebras are presented.