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 G2 is completely solved.