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Products, crossed products, and Zappa--Szép products for $k$-graphs

Adlin Abell-Ball, Elizabeth Gillaspy, George Glidden-Handgis, S. Joseph Lippert

TL;DR

The paper develops a Zappa--Szép framework to connect product graphs with $k$-graphs and their $C^*$-algebras by introducing quasi-products and stability notions. It shows that when one action between factors is trivial, the $C^*$-algebra of a quasi-product decomposes as a crossed product or, under favorable conditions, as a tensor product; it also provides graph-theoretic criteria for when a quasi-product is actually a genuine product. Key contributions include a precise decomposition of $(k_1+k_2)$-graphs into matched pairs, a crossed-product decomposition for cycle-like factors, and new stability concepts (stable and relaxed stable quasi-factors) that yield product decompositions in broad cases, including polytrees and $k$-trees. This framework clarifies when product-like $C^*$-algebras arise from more general quasi-product structures and offers practical criteria for identifying such decompositions in examples.

Abstract

We use the lens of Zappa--Szép decomposition to examine the relationship between directed graph products and $k$-graph products. There are many examples of higher-rank graphs, or $k$-graphs, whose underlying directed graph may be factored as a product, but the $k$-graph itself is not a product. In such examples, we establish that the Zappa--Szép structure of the $k$-graph gives rise to "actions'' of the underlying directed factors on each other. Although these "actions'' are in general poorly behaved, if one of them is trivial (or trivial up to isomorphism), we obtain a crossed-product-like structure on the $k$-graph. We provide examples where this crossed-product structure is visible in the associated $C^*$-algebra, and we characterize those $k$-graphs whose Zappa--Szép induced actions are trivial up to isomorphism.

Products, crossed products, and Zappa--Szép products for $k$-graphs

TL;DR

The paper develops a Zappa--Szép framework to connect product graphs with -graphs and their -algebras by introducing quasi-products and stability notions. It shows that when one action between factors is trivial, the -algebra of a quasi-product decomposes as a crossed product or, under favorable conditions, as a tensor product; it also provides graph-theoretic criteria for when a quasi-product is actually a genuine product. Key contributions include a precise decomposition of -graphs into matched pairs, a crossed-product decomposition for cycle-like factors, and new stability concepts (stable and relaxed stable quasi-factors) that yield product decompositions in broad cases, including polytrees and -trees. This framework clarifies when product-like -algebras arise from more general quasi-product structures and offers practical criteria for identifying such decompositions in examples.

Abstract

We use the lens of Zappa--Szép decomposition to examine the relationship between directed graph products and -graph products. There are many examples of higher-rank graphs, or -graphs, whose underlying directed graph may be factored as a product, but the -graph itself is not a product. In such examples, we establish that the Zappa--Szép structure of the -graph gives rise to "actions'' of the underlying directed factors on each other. Although these "actions'' are in general poorly behaved, if one of them is trivial (or trivial up to isomorphism), we obtain a crossed-product-like structure on the -graph. We provide examples where this crossed-product structure is visible in the associated -algebra, and we characterize those -graphs whose Zappa--Szép induced actions are trivial up to isomorphism.

Paper Structure

This paper contains 10 sections, 16 theorems, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Higher-rank graphs
  4. Zappa--Szép products
  5. Quasi-products
  6. Cycle Graph Factors
  7. Stability and products
  8. Motivating examples
  9. Relaxed stability and products
  10. Applications: polytrees and $k$-trees

Key Result

Theorem 2.4

efgggp Let $G$ be an edge-colored directed graph. If $\sim$ is any $(s,r, d)$-preserving equivalence relation $\sim$ on the path category $G^*$ of an edge-colored directed graph $G$, then $G^*/\sim$ is a $k$-graph iff $\sim$ satisfies the conditions (KG0) and (KG4) below.

Figures (1)

  • Figure 4.3: Above is pictured $C_{4,3}$. Note that the factorization rule is uniquely determined. This trait persists for arbitrary $C_{n,k_2}$.

Theorems & Definitions (55)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Example 2.5
  • Remark 2.6
  • Example 2.7
  • Definition 2.8
  • Lemma 2.9
  • Definition 3.1
  • ...and 45 more