Determinantal formulae for the Casimir operators of inhomogeneous Lie algebras

TL;DR

This paper develops matrix formulas for Casimir operators of inhomogeneous Lie algebras using contractions and Gel'fand's method, extending to various algebra types and providing new analytical tools.

Contribution

It introduces a systematic matrix approach for Casimir operators of inhomogeneous Lie algebras, including extensions and contractions, enhancing invariant computation methods.

Findings

Matrix formulae for Casimir operators of inhomogeneous pseudo-unitary algebras.

Extension of the method to algebras isomorphic to extensions by derivations.

Application to other 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.