Leavitt path algebras with finitely presented irreducible representations

TL;DR

This paper characterizes when all irreducible representations of Leavitt path algebras are finitely presented, linking algebraic and graphical conditions, and explores the relationship with Gelfand-Kirillov dimension for finite and infinite graphs.

Contribution

It provides necessary and sufficient algebraic and graphical conditions for finite presentation of irreducible representations in Leavitt path algebras, extending understanding to infinite graphs.

Findings

Graph E must be row finite for all irreducible representations to be finitely presented.

Cycles in E form an artinian partial order under a specific preorder.

Finite graphs with these properties have finite Gelfand-Kirillov dimension, but this does not extend to infinite graphs.

Abstract

Let E be an arbitrary graph, K be any field and let L be the corresponding Leavitt path algebra. Necessary and sufficient conditions (which are both algebraic and graphical) are given under which all the irreducible representations of L are finitely presented. In this case, the graph E turns out to be row finite and the cycles in E form an artinian partial ordered set under a defined preorder. When the graph E is finite, the above graphical conditions were shown to be equivalent to the algebra L having finite Gelfand-Kirillov dimension in a paper by Alahmadi, Alsulami, Jain and Zelmanov. Examples are constructed showing that this equivalence no longer holds if the graph is infinite and a complete description is obtained of Leavitt path algebras over arbitrary graphs having finite Gelfand-Kirillov dimension