Maximal tails,character fibres and induced modules for pullback Kumjian-Pask algebras

Abstract

Let $f:{\mathbb{N}}^k\to {\mathbb{N}}^{\ell }$ be a surjective monoid homomorphism and let $\Gamma$ be a row-finite $\ell$-graph with no sources and finitely many vertices. We give an explicit graded isomorphism from the Kumjian--Pask algebra of the pullback $f^{\ast }\Gamma$ onto the tensor product of $\KP_{\K}(Γ)$ and the group algebra of the kernel of the group completion of $f$. When $\Gamma$ is strongly aperiodic, but need not be cofinal, every maximal tail $T$ and every maximal ideal $\mathfrak{m}$ of the kernel group algebra determine an explicit primitive ideal and primitive quotient. If, in addition, $\K$ is uncountable and algebraically closed, these ideals exhaust the primitive spectrum. We prove that the resulting parametrisation is a homeomorphism for the product of the maximal-tail and Zariski topologies. Each primitive ideal is realised as the annihilator of a simple module induced from the isotropy of a path which is cofinal in $T$, and the character fibres are algebraic tori. Two examples exhibit, respectively, a single character fibre and the non-Hausdorff gluing of two such fibres.