On graded irreducible representations of Leavitt path algebras

TL;DR

This paper explores the structure of graded irreducible representations of Leavitt path algebras, introducing concepts like Laurent vertices and characterizing when all representations are finitely presented, with implications for algebraic regularity.

Contribution

It provides a detailed classification of graded irreducible representations and characterizes Leavitt path algebras with all graded ideals and representations, introducing the concept of Laurent vertices.

Findings

Minimal graded left ideals are generated by Laurent vertices and line points.

A representation is graded iff associated path is irrational infinite.

All one-sided ideals are graded iff the graph has no cycles.

Abstract

Using the E-algebraic systems, various graded irreducible representations of a Leavitt path algebra L of a graph E over a field K are constructed. The concept of a Laurent vertex is introduced and it is shown that the minimal graded left ideals of L are generated by Laurent vertices and/or line points leading to a detailed description of the graded socle of L. Following this, a complete characterization is obtained of the Leavitt path algebras over which every graded irreducible representation is finitely presented. A useful result is that the irreducible representation V_[p] induced by infinite path tail-equivalent to an infinite path p is graded if and only if p is an irrational infinite path. We also show that every one-sided ideal of L is graded if and only if the graph E contains no cycles. Supplementing a theorem of one of the co-authors that every Leavitt path algebra is graded von Neumann regular, we describe the graded self-injective Leavitt path algebras. These turn out to be direct sums of matrix rings of arbitrary size over K and k[x,x^-1].