Iterating the Cuntz-Nica-Pimsner construction for compactly aligned product systems

In this article we study how decompositions of a quasi-lattice ordered group $(G,P)$ relate to decompositions of the Nica-Toeplitz algebra $\mathcal{NT}_\mathbf{X}$ and Cuntz-Nica-Pimsner algebra $\mathcal{NO}_\mathbf{X}$ of a compactly aligned product system $\mathbf{X}$ over $P$. In particular, we are interested in the situation where $(G,P)$ may be realised as the semidirect product of quasi-lattice ordered groups. Our results generalise Deaconu's work on iterated Toeplitz and Cuntz-Pimsner algebras - we show that the Nica-Toeplitz algebra and Cuntz-Nica-Pimsner algebra of a compactly aligned product system over $\mathbb{N}^k$ may be realised as $k$-times iterated Toeplitz and Cuntz-Pimsner algebras respectively.