On reflexive simple modules in Artin algebras

TL;DR

This paper investigates whether the selfinjectivity of an Artin algebra can be characterized by the reflexivity of its simple modules, providing positive results for broad classes including Gorenstein and QF-3 algebras.

Contribution

It introduces a new characterization of selfinjective Artin algebras based on the reflexivity of simple modules, expanding understanding of algebraic properties.

Findings

Selfinjectivity characterized by simple module reflexivity in certain classes

Positive results for Gorenstein and QF-3 algebras

Provides motivation and framework for further research

Abstract

Let $A$ be an Artin algebra. It is well known that $A$ is selfinjective if and only if every finitely generated $A$-module is reflexive. In this article we pose and motivate the question whether an algebra $A$ is selfinjective if and only if every simple module is reflexive. We give a positive answer to this question for large classes of algebras which include for example all Gorenstein algebras and all QF-3 algebras.