Leavitt Path Algebras as Flat Bimorphic Localizations

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.