The Groupoid Approach to Leavitt Path Algebras

Abstract

When the theory of Leavitt path algebras was already quite advanced, it was discovered that some of the more difficult questions were susceptible to a new approach using topological groupoids. The main result that makes this possible is that the Leavitt path algebra of a graph is graded isomorphic to the Steinberg algebra of the graph's boundary path groupoid.This expository paper has three parts: Part 1 is on the Steinberg algebra of a groupoid, Part 2 is on the path space and boundary path groupoid of a graph, and Part 3 is on the Leavitt path algebra of a graph. It is a self-contained reference on these topics, intended to be useful to beginners and experts alike. While revisiting the fundamentals, we prove some results in greater generality than can be found elsewhere, including the uniqueness theorems for Leavitt path algebras.

A classification of ideals in Steinberg and Leavitt path algebras over arbitrary rings

“…. This map can be extended uniquely to an R-algebra homomorphism L R (E) → A R (G E ), and [14,Theorem 2.7] shows that this is an isomorphism. In particular, A R (G E ) is generated as an R-module by the set…”

“…Leavitt path algebras are algebras associated to directed graphs, and are a special kind of Steinberg algebras. We will give a brief introduction on their definition and how they become Steinberg algebras, but we refer to [1,14] and the introduction for more information, applications, and motivations. The purpose of this section is to specialise the contents of §3 to boundary path groupoids, moving from a topological setting towards a more combinatorial setting and simplifying some aspects of the theory where it is possible to do so.…”

“…As with ∂E, we give G E the topology generated by such sets Z(α, β, F ). For this topology, G E is a Hausdorff ample groupoid and each set Z(α, β, F ) is a compact open bisection [14,Theorem 2.4]. Note that for each α ∈ E * and F ⊆ finite s −1 (r(α)), we have…”

Tensor Products of Steinberg Algebras

“…In short, L n,R is the Leavitt path algebra of the graph with one vertex and n edges, and O n is the boundary path groupoid of the same graph. It is well-known (see [22,31]) that A R (O n ) ∼ = L n,R , and the isomorphism restricts to D R (O n ) ∼ = D n,R . Complete, classical definitions are given below.…”

“…We claim that for each f ∈ A R (G), there exists a finite set F ⊆ B whose elements are mutually disjoint, such that f is a linear combination of the set {1 B | B ∈ F }. By[31, Corollary 1.14], there is an expression f Di where s 1 , . .…”