Simple reflexive modules over finite-dimensional algebras

Abstract

Let A be a finite-dimensional algebra. If A is self-injective, then all modules are reflexive. Marczinzik recently has asked whether A has to be self-injective in case all the simple modules are reflexive. Here, we exhibit an 8-dimensional algebra which is not self-injective, but such that all simple modules are reflexive (actually, for this example, the simple modules are the only non-projective indecomposable modules which are reflexive). In addition, we present some properties of simple reflexive modules in general. Marczinzik had motivated his question by providing large classes of algebras such that any algebra in the class which is not self-injective has simple modules which are not reflexive. However, as it turns out, most of these classes have the property that any algebra in the class which is not self-injective has simple modules which are not even torsionless.