Twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs

TL;DR

This paper introduces twisted relative Cuntz-Krieger algebras for finitely aligned higher-rank graphs, establishing their properties, ideal structure, and isomorphisms, extending the theory of graph algebras with cocycle twists.

Contribution

It constructs and analyzes twisted relative Cuntz-Krieger algebras for higher-rank graphs, including gauge actions, uniqueness theorems, and ideal structure descriptions.

Findings

Existence of gauge actions on these algebras

Gauge-invariant uniqueness theorem established

Complete graph-theoretic description of ideal structure

Abstract

To each finitely aligned higher-rank graph $\Lambda$ and each $\mathbb{T}$-valued 2-cocycle on $\Lambda$, we associate a family of twisted relative Cuntz-Krieger algebras. We show that each of these algebras carries a gauge action, and prove a gauge-invariant uniqueness theorem. We describe an isomorphism between the fixed point algebras for the gauge actions on the twisted and untwisted relative Cuntz-Krieger algebras. We show that the quotient of a twisted relative Cuntz-Krieger algebra by a gauge-invariant ideal is canonically isomorphic to a twisted relative Cuntz-Krieger algebra associated to a subgraph. We use this to provide a complete graph-theoretic description of the gauge-invariant ideal structure of each twisted relative Cuntz-Krieger algebra.