Products, crossed products, and Zappa--Sz\'ep products for $k$-graphs

Abstract

We use the lens of Zappa--Sz\'ep 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--Sz\'ep 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--Sz\'ep induced actions are trivial up to isomorphism.