Cyclic adjoint modules and their embeddings in quantized enveloping algebras

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.