Indecomposable finite-dimensional representations of a class of Lie algebras and Lie superalgebras

TL;DR

This paper classifies indecomposable finite-dimensional representations of certain Lie algebras and superalgebras using universal enveloping algebra methods, revealing new low-dimensional representations and classifying ideals in specific cases.

Contribution

It provides a complete classification of ideals in the enveloping algebra of the translation subgroup and constructs new low-dimensional indecomposable representations.

Findings

Classified ideals in the enveloping algebra of the translation subgroup.

Constructed a 12-dimensional family of inequivalent representations.

Identified complexities in supersymmetric cases.

Abstract

In the article at hand, we sketch how, by utilizing nilpotency to its fullest extent (Engel, Super Engel) while using methods from the theory of universal enveloping algebras, a complete description of the indecomposable representations may be reached. In practice, the combinatorics is still formidable, though. It turns out that the method applies to both a class of ordinary Lie algebras and to a similar class of Lie superalgebras. Besides some examples, due to the level of complexity we will only describe a few precise results. One of these is a complete classification of which ideals can occur in the enveloping algebra of the translation subgroup of the Poincar\'e group. Equivalently, this determines all indecomposable representations with a single, 1-dimensional source. Another result is the construction of an infinite-dimensional family of inequivalent representations already in dimension 12. This is much lower than the 24-dimensional representations which were thought to be the lowest possible. The complexity increases considerably, though yet in a manageable fashion, in the supersymmetric setting. Besides a few examples, only a subclass of ideals of the enveloping algebra of the super Poincar\'e algebra will be determined in the present article.