Decomposition of Enveloping Algebras of Simple Lie Algebras and their Related Polynomial Algebras

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 case of rank-two simple Lie algebras is revisited and completed with the analysis of the exceptional Lie algebra G 2 .