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

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.