The basics of Leavitt path algebras: Motivations, definitions and examples

Abstract

We introduce the central idea, that of a Leavitt path algebra. We start by describing the classical Leavitt algebras. We then proceed to give the definition of the Leavitt path algebra LK(E) for an arbitrary directed graph E and field K. After providing some basic examples, we show how Leavitt path algebras are related to the monoid realization algebras of Bergman, as well as to graph C∗-algebras. We then introduce the more general construction of relative Cohn path algebras CKX(E), and show how these are related to Leavitt path algebras. We finish by describing how any Cohn (specifically, Leavitt) path algebra may be constructed as a direct limit of Cohn (specifically, Leavitt) path algebras corresponding to finite graphs. We conclude the chapter with an historical overview of the subject.