What question did this study set out to answer?

The aim is to address the decomposition of enveloping algebras of simple Lie algebras, focusing on their relationship with nilpotent subalgebras.

March 29, 2026Open Access

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

Key Points

The aim is to address the decomposition of enveloping algebras of simple Lie algebras, focusing on their relationship with nilpotent subalgebras.
Combining analytical and algebraic approaches to study enveloping algebras.
Investigating the internal labelling problem related to nilpotent subalgebras.
Revisiting and completing the analysis of rank-two simple Lie algebras with special attention to exceptional Lie algebra G2.
Establishing a lower bound for the number of generators of the commutant.
Identifying the structure of the maximal abelian subalgebra.
Providing a comprehensive understanding of the case for rank-two simple Lie 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 case of rank-two simple Lie algebras is revisited and completed with the analysis of the exceptional Lie algebra G 2 .

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