Decomposition of enveloping algebras of simple Lie algebras and their related polynomial algebras

TL;DR

This paper investigates the decomposition of enveloping algebras of simple Lie algebras, linking it to polynomial algebras and solving the case for the exceptional Lie algebra G2, providing bounds on generators and subalgebras.

Contribution

It introduces a combined analytical and algebraic approach to the decomposition problem, including explicit solutions for G2 and bounds for generators.

Findings

Complete solution for the decomposition of G2 enveloping algebra

Lower bounds for the number of generators of the commutant

Bounds for maximal Abelian subalgebras

Abstract

The decomposition problem of the enveloping algebra of a simple Lie algebra is reconsidered combining both the analytical and the algebraic approach, showing its relation with the internal labelling problem with respect to a nilpotent subalgebra. A lower bound for the number of generators of the commutant as well as the maximal Abelian subalgebra are obtained. The decomposition problem for the exceptional Lie algebra $G_2$ is completely solved.