Classification of the Reducible Verma Modules over the Jacobi Algebra $ {\cal G}_2$

TL;DR

This paper classifies reducible Verma modules over the Jacobi algebra ${\cal G}_2$, analyzing their reducibility, singular vectors, and embedding patterns to provide a comprehensive understanding of their structure.

Contribution

It introduces a systematic classification of reducible Verma modules over the Jacobi algebra, including explicit constructions and embedding diagrams.

Findings

Complete classification of reducible Verma modules

Explicit construction of singular vectors

Embedding diagrams illustrating module relations

Abstract

In the present paper we study the representations of the Jacobi algebra. More concretely, we define, analogously to the case of semi-simple Lie algebras, the Verma modules over the Jacobi algebra ${\cal G}_2$. We study their reducibility and give explicit construction of the reducible Verma modules exhibiting the corresponding singular vectors. Using this information we give a complete classification of the reducible Verma modules. More than this we exhibit their interrelation of embeddings between these modules. These embeddings are illustrated by diagrams of the embedding patterns so that each reducible Verma module appears in one such diagram.