Cyclic adjoint modules and their embeddings in quantized enveloping algebras

TL;DR

This paper investigates cyclic adjoint modules in quantized enveloping algebras, classifies their embeddings, and explores their structure and minimal elements, revealing infinite families and finite generation properties.

Contribution

It introduces a classification of embeddings of irreducible modules via cyclic generators and analyzes the structure of cyclic adjoint modules in quantum groups.

Findings

Classified embeddings of finite-dimensional irreducible modules in quantized enveloping algebras.

Identified infinite families of realizations in the cominuscule case.

Proved finite generation of cyclic adjoint modules by irreducible submodules.

Abstract

We study cyclic adjoint modules arising from the relative locally finite part of the adjoint action of a quantum Levi subalgebra on a quantized enveloping algebra. We classify embeddings of finite-dimensional irreducible modules inside of quantized enveloping algebra via cyclic generators and show that such realizations are in general non-unique, exhibiting infinite families in the cominuscule case. We also introduce a partial order on cyclic adjoint modules, characterize its minimal elements, and prove finite generation by irreducible submodules.