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

Rutwig Campoamor-Stursberg, Ian Marquette

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.

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 is completely solved.
Paper Structure (10 sections, 1 theorem, 70 equations, 3 tables)

This paper contains 10 sections, 1 theorem, 70 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Presentation of semisimple Lie algebras
  3. Decomposition of enveloping algebras and commutants
  4. Analytic properties of the commutant
  5. Analysis of rank $2$ semisimple Lie algebras
  6. $D_2=so(4)$
  7. $A_2=\mathfrak{sl}(3,\mathbb{C})$
  8. $B_2=\mathfrak{so}(5,\mathbb{C})$
  9. The exceptional Lie algebra $G_2$
  10. Conclusions

Key Result

Proposition 1

Let $\mathfrak{n}$ be the nilradical of the Borel subalgebra of a complex simple Lie algebra $\mathfrak{s}$.

Theorems & Definitions (1)

  • Proposition 1