Injective Modules over Down-Up Algebras

TL;DR

This paper investigates the conditions under which injective hulls of simple modules over Noetherian Down-Up algebras are finite or locally Artinian, revealing a precise characterization related to algebraic properties.

Contribution

It characterizes fully bounded Noetherian Down-Up algebras as those module-finite over a central subalgebra and describes conditions for injective hulls to be locally Artinian.

Findings

Fully bounded Noetherian Down-Up algebras are module-finite over a central subalgebra.

Injective hulls of simple modules are locally Artinian under specific root conditions.

Provides a classification linking algebraic roots to module-theoretic properties.

Abstract

The purpose of this paper is to study finiteness conditions on injective hulls of simple modules over Noetherian Down-Up algebras. We will show that the Noetherian Down-Up algebras A(\alpha,\beta,\gamma) which are fully bounded are precisely those which are module-finite over a central subalgebra. We show that injective hulls of simple A(\alpha,\beta,\gamma)-modules are locally Artinian provided the roots of X^2-\alpha X-\beta are distinct roots of unity or both equal to one.