Moves on $k$-graphs preserving Morita equivalence

TL;DR

This paper extends the geometric classification of graph $C^*$-algebras to higher-rank $k$-graphs by identifying four moves that preserve Morita equivalence, linking $k$-graphs to their underlying directed graphs.

Contribution

It introduces four moves for $k$-graphs that preserve Morita equivalence and connects $k$-graphs with their underlying directed graphs, expanding classification methods.

Findings

Identified four moves: insplitting, delay, sink deletion, reduction.

Proved these moves preserve Morita equivalence of $C^*$-algebras.

Established new results relating $k$-graphs to their underlying directed graphs.

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 $\Lambda$, which leave invariant the Morita equivalence class of its $C^*$-algebra $C^*(\Lambda)$. 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.