Leavitt path algebras and their representations

TL;DR

This paper explores the representation theory of Leavitt path algebras derived from finite and infinite digraphs, highlighting the impact of infinite emitters and connecting to broader algebraic structures.

Contribution

It introduces a new approach to constructing and analyzing representations of Leavitt path algebras using ideal topology and extends understanding to infinite emitters.

Findings

Representations of Leavitt path algebras can be induced from quiver algebra representations.

Infinite emitters significantly influence the structure and extensions of representations.

Connections established between Leavitt path algebras, localizations, quiver algebras, and noncommutative domains.

Abstract

Viewing Leavitt path algebras of finite digraphs as rings of quotients defined by the ideal topology of the ideal generated by all arrows and sinks allows us to induce their representations from those of the quiver algebras and therefore provides a way to construct representations of Leavitt path algebras of not necessarily finite digraphs together with a computation of the endomorphism rings. This approach emphasizes the decisive role of infinite emitters, i.e., vertices with infinitely many outgoing arrows, in the representation theory of Leavitt path algebras. In particular, extensions of representations of ordinary quivers by taking tensor products are no longer simple extensions when infinite emitters exist, hence they become the targets of further study. Our results connect Leavitt path algebras to other vigorously active working areas in ring theory like localizations, quiver algebras, free associative algebras as well as noncommutative noetherian domains. Moreover, as Cuntz algebras ${\mathcal O}_n$ for operator graph algebras, our treatment emphasizes the central role of classical Leavitt algebras $L_K(1, n)$ in the study of Leavitt path algebras.