Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators

TL;DR

This paper constructs polynomial operators in the enveloping algebra of semidirect product Lie algebras that mimic semisimple parts and facilitate explicit computation of Casimir operators, with applications to Hamilton algebras.

Contribution

It introduces the concept of virtual copies of semisimple Lie algebras within enveloping algebras to compute Casimir operators explicitly.

Findings

Constructed polynomial operators commuting with solvable parts

Derived closed-form Casimir operators for semidirect products

Analyzed virtual copies under Lie algebra contractions

Abstract

Given a semidirect product $\frak{g}=\frak{s}\uplus\frak{r}$ of semisimple Lie algebras $\frak{s}$ and solvable algebras $\frak{r}$, we construct polynomial operators in the enveloping algebra $\mathcal{U}(\frak{g})$ of $\frak{g}$ that commute with $\frak{r}$ and transform like the generators of $\frak{s}$, up to a functional factor that turns out to be a Casimir operator of $\frak{r}$. Such operators are said to generate a virtual copy of $\frak{s}$ in $\mathcal{U}(\frak{g})$, and allow to compute the Casimir operators of $\frak{g}$ in closed form, using the classical formulae for the invariants of $\frak{s}$. The behavior of virtual copies with respect to contractions of Lie algebras is analyzed. Applications to the class of Hamilton algebras and their inhomogeneous extensions are given.