Abstract
The algebraic structures known as Leavitt path algebras were initially developed in 2004 by Ara, Moreno and Pardo, and almost simultaneously (using a different approach) by the author and Aranda Pino. During the intervening decade, these algebras have attracted significant interest and attention, not only from ring theorists, but from analysts working in C -algebras, group theorists, and symbolic dynamicists as well. The goal of this article is threefold: to introduce the notion of Leavitt path algebras to the general mathematical community; to present some of the important results in the subject; and to describe some of the field’s currently unresolved questions.
Highlights
Our goal in writing this article is threefold: first, to provide a history and overall viewpoint of the ideas which comprise the subject of Leavitt path algebras; second, to give the reader a general sense of the results which have been achieved in the field; and to give a broad picture of some of the research lines which are currently being pursued
With the purely infinite simple graph C∗-algebras as motivation, the four authors in [29] introduced the “algebraic Cuntz–Krieger (CK) algebras.” (Retrospectively, these are seen to be the Leavitt path algebras corresponding to finite graphs having neither sources nor sinks, and which do not consist of a disjoint union of cycles.) These algebraic Cuntz–Krieger algebras arose as specific examples of fractional skew monoid rings, and the germane ones were shown to be purely infinite simple by using techniques which applied to the more general class
We conclude our discussion of Historical Plot Line #1 in the development of Leavitt path algebras by again quoting Enrique Pardo: For us the motivation was to give an algebraic framework to all these families of C∗-algebras associated to combinatorial objects, say Cuntz–Krieger algebras and graph C∗-algebras
Summary
The fundamental examples of rings that are encountered during one’s algebraic pubescence (e.g., fields K , Z, K [x], K [x, x−1], Mn(K )) all have the following property. With the purely infinite simple graph C∗-algebras as motivation, the four authors in [29] introduced the “algebraic Cuntz–Krieger (CK) algebras.” (Retrospectively, these are seen to be the Leavitt path algebras corresponding to finite graphs having neither sources nor sinks, and which do not consist of a disjoint union of cycles.) These algebraic Cuntz–Krieger algebras arose as specific examples of fractional skew monoid rings, and the germane ones were shown to be purely infinite simple by using techniques which applied to the more general class. We conclude our discussion of Historical Plot Line #1 in the development of Leavitt path algebras by again quoting Enrique Pardo: For us the motivation was to give an algebraic framework to all these families of (purely infinite simple) C∗-algebras associated to combinatorial objects, say Cuntz–Krieger algebras and graph C∗-algebras For this reason we always looked at properties that were known in C∗ case and were related to combinatorial information: we wanted to know which part of these results relies in algebraic information, and which ones in analytic information. L K (E) is simple if and only if the only hereditary saturated subsets of E are trivial, and every cycle in E has an exit
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