Dynkin abelianisations of Lie algebras of exceptional type

TL;DR

This paper explores degenerations of exceptional Lie algebras via affine embeddings, establishing a filtration that leads to abelianisations and revealing that their associated graded modules are Demazure modules.

Contribution

It introduces a novel filtration approach to degenerate exceptional Lie algebras and connects their modules to Demazure modules, advancing understanding of their structure.

Findings

Associated graded modules are isomorphic to Demazure modules.

Filtration provides a new perspective on degenerations of Lie algebras.

Method applies to all simple Lie algebras of exceptional type.

Abstract

We consider degenerations of all simple Lie algebras of exceptional type obtained by embedding into affine Lie algebras. We give a filtration to consider this as an abelianisation of the original Lie algebra. We then show that the associated graded of simple, finite-dimensional modules are isomorphic to Demazure modules.