Injectives over Leavitt path algebras of graphs with disjoint cycles

TL;DR

This paper explicitly constructs the injective envelopes of simple modules over Leavitt path algebras of finite graphs with at most one cycle per vertex, using a formal power series extension approach.

Contribution

It provides a new explicit construction of injective envelopes for simple modules over Leavitt path algebras of graphs satisfying Condition (AR).

Findings

Injective envelopes are constructed explicitly for these algebras.

The method extends previous understanding from Toeplitz algebra to more general graphs.

The approach uses a formal power series extension of modules.

Abstract

Let $K$ be any field, and let $E$ be a finite graph with the property that every vertex in $E$ is the base of at most one cycle (we say such a graph satisfies Condition (AR)). We explicitly construct the injective envelope of each simple left module over the Leavitt path algebra $L_K(E)$. The main idea girding our construction is that of a "formal power series" extension of modules, thereby developing for all graphs satisfying Condition (AR) the understanding of injective envelopes of simple modules over $L_K(E)$ achieved previously for the simple modules over the Toeplitz algebra.