Casimir operators induced by Maurer-Cartan equations

TL;DR

This paper demonstrates how to explicitly construct Casimir operators for certain inhomogeneous Lie algebras using Maurer-Cartan equations, establishing bounds on representation dimensions and extending to rational invariants.

Contribution

It introduces a method to derive Casimir operators from Maurer-Cartan equations for specific Lie algebras, providing explicit constructions and bounds.

Findings

Casimir operators can be constructed from Maurer-Cartan equations for certain Lie algebras.

The method imposes bounds on the dimension of the algebra's representations.

The approach is extended to compute rational invariants of some Lie algebras.

Abstract

It is shown that for inhomogeneous Lie algebras $\frak{g}=\frak{s}\overrightarrow{\oplus}_{\Lambda}(\dim \Lambda)L_{1}$ satisfying the condition $\mathcal{N}(\frak{g})=1$, the only Casimir operator can be explicitly constructed from the Maurer-Cartan equations by means of wedge products. It is shown that this constraint imposes sharp bounds for the dimension of the representation $R$. The procedure is generalized to compute also the rational invariant of some Lie algebras.