Coverings of k-graphs

Abstract

k-graphs are higher-rank analogues of directed graphs which were first developed to provide combinatorial models for operator algebras of Cuntz-Krieger type. Here we develop the theory of covering spaces for k-graphs, obtaining a satisfactory version of the usual topological classification in terms of subgroups of a fundamental group. We then use this classification to describe the C * -algebras of covering k-graphs as crossed products by coactions of homogeneous spaces, generalizing recent results on the C * -algebras of graphs.  2005 Elsevier Inc. All rights reserved.

Disjoint Sets in Graphs and its Application to Electrical Networks

“…This monitoring problem was introduced and studied in (Pask et al, 2005;Katrenic and Semanisin, 2010).…”

Real rank and topological dimension of higher rank graph algebras

“…Following [21,25,32] we briefly recall the notion of a k-graph and the associated notation. For k ≥ 0, a k-graph is a nonempty countable small category equipped with a functor d : Λ → N k that satisfies the factorisation property: for all λ ∈ Λ and m, n ∈ N k such that d(λ) = m + n there exist unique µ, ν ∈ Λ such that d(µ) = m, d(ν) = n, and λ = µν.…”

Generalised morphisms of 𝑘-graphs: 𝑘-morphs

“…Modulo the minor differences mentioned above, we will adopt the conventions of [18,26] for k-graphs. Given a nonnegative integer k, a k-graph is a nonempty countable small category Λ equipped with a functor d : Λ → N k satisfying the factorisation property: for all λ ∈ Λ and m, n ∈ N k such that d(λ) = m + n, there exist unique μ, ν ∈ Λ such that d(μ) = m, d(ν) = n, and λ = μν.…”

“…As in [26], a covering of a k-graph Λ by a k-graph Γ is a surjective k-graph morphism p : Γ → Λ such that for all v ∈ Γ 0 , p restricts to bijections between vΓ and p(v)Λ and between Γv and Λp(v). The covering p : Γ → Λ is finite if p −1 (v) is finite for all v ∈ Λ 0 .…”