A New Geometric Morita Invariant for Higher Rank Graph $C^*$-algebras

TL;DR

This paper introduces a new geometric move called LiMaR-split for higher rank graphs ($k$-graphs) and demonstrates its preservation of $C^*$-algebras up to Morita equivalence, advancing the classification of these algebras.

Contribution

It proposes the LiMaR-split move for $k$-graphs and proves it preserves the associated $C^*$-algebras up to Morita equivalence, extending geometric classification methods.

Findings

LiMaR-split preserves $k$-graph $C^*$-algebras under certain conditions

The move generalizes the outsplit move for directed graphs

Advances the geometric classification program for $k$-graph $C^*$-algebras

Abstract

Higher rank graphs, also known as $k$-graphs, are a $k$-dimensional generalization of directed graphs and a rich source of examples of $C^*$-algebras. In the present paper, we contribute to the geometric classification program for $k$-graph $C^*$-algebras by introducing a new move on $k$-graphs, called LiMaR-split, which is a generalization of outsplit for directed graphs. We show, under one additional assumption, that LiMaR-split preserves the $k$-graph $C^*$-algebras up to Morita equivalence.