Leavitt Path Algebras as Flat Bimorphic Localizations

TL;DR

This paper demonstrates that Leavitt path algebras can be understood as flat bimorphic localizations of free associative algebras, connecting them to localization theory and providing new insights into their structure and module properties.

Contribution

It provides a novel, conceptual description of Leavitt algebras as rings of quotients via perfect Gabriel topologies, linking them to localization theory and automorphism groups.

Findings

Leavitt algebras are rings of right quotients of free associative algebras.

Finite graph Leavitt path algebras are quotients of quiver algebras via perfect Gabriel topology.

The Toeplitz algebra can be realized as a flat ring of quotients.

Abstract

Refining an idea of Rosenmann and Rosset we show that the now widely studied classical Leavitt algebra $L_K(1,n)$ over a field $K$ is a ring of right quotients of the unital free associative algebra of rank $n$ with respect to the perfect Gabriel topology defined by powers of an ideal of codimension 1, providing a conceptual, variable-free description of $L_K(1, n)$. This result puts Leavitt (path) algebras on the frontier of important research areas in localization theory, free ideal rings and their automorphism groups, quiver algebras and graph operator algebras. As applications one obtains a short, transparent proof for the module type $(1,n)\, (n\geq 2)$ of Leavitt algebra $L_K(1, n)\, (n\geq 2)$, and the fact that Leavitt path algebras of finite graphs are rings of quotients of corresponding ordinary quiver algebras with respect to the perfect Gabriel topology defined by powers of the ideal generated by all arrows and sinks. In particular, the Jacobson algebra of one-sided inverses, that is, the Toeplitz algebra, can also be realized as a flat ring of quotients, further illuminating the rich structure of these beautiful, useful algebras.