Cyclic adjoint modules and their embeddings in quantized enveloping algebras

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 analyze the realization of irreducible modules inside the quantized enveloping algebra via cyclic generators and describe embeddings of a fixed type. This leads to a natural map to isomorphism classes, whose fibers reflect the non-uniqueness of such realizations. We further introduce a partial order on cyclic adjoint modules and relate its minimal elements to irreducible submodules. In addition, we show that every cyclic adjoint module is generated by finitely many irreducible submodules.

Theorems & Definitions (49)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 1
• proof
• Lemma 2
• proof
• Definition 5
• Proposition 6