Indecomposable Injectives over the Jacobson Algebra

TL;DR

This paper classifies all indecomposable left injective modules over the Jacobson algebra, a specific algebra defined by generators and a relation, extending previous work on injective envelopes of simple modules.

Contribution

It provides a complete list of indecomposable injective modules over the Jacobson algebra, advancing understanding of its module category.

Findings

Complete classification of indecomposable injectives

Extension of previous results on injective envelopes

Deeper insight into the structure of the Jacobson algebra

Abstract

Let K be any field. In this paper we give a complete list of the indecomposable left injective module over the Jacobson algebra K<X,Y | XY = 1>, i.e., the free associative K-algebra on two (noncommuting) generators, modulo the single relation XY = 1. This is the natural continuation of the paper of the second two authors with Gene Abrams on the charaterization of the injective envelope of the simple modules over K<X, Y | XY = 1>.