Moves on k-graphs preserving Morita equivalence

2020/06/24 by Caleb Eckhardt, Eckhardt, Caleb, Kit Fieldhouse +9 · 2 citations
Mathematics · #37B51 #46L05 #46L35 #46L55 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2006.13441

openalex publication_date 2020/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We initiate the program of extending to higher-rank graphs (k-graphs) the geometric classification of directed graph C^*-algebras, as completed in the 2016 paper of Eilers, Restorff, Ruiz, and Sorensen [ERRS16]. To be precise, we identify four "moves," or modifications, one can perform on a k-graph Λ, which leave invariant the Morita equivalence class of its C^*-algebra C^*(Λ). These moves -- insplitting, delay, sink deletion, and reduction -- are inspired by the moves for directed graphs described by Sorensen [S\o13] and Bates-Pask [BP04]. Because of this, our perspective on k-graphs focuses on the underlying directed graph. We consequently include two new results, Theorem 2.3 and Lemma 2.9, about the relationship between a k-graph and its underlying directed graph.

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• Higher-rank graphs and the graded K-theory of Kumjian-Pask Algebras
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