Maximal tails,character fibres and induced modules for pullback Kumjian-Pask algebras
Abstract
Let f:Nk→Nℓ be a surjective monoid homomorphism and let Γ be a row-finite ℓ-graph with no sources and finitely many vertices. We give an explicit graded isomorphism from the Kumjian--Pask algebra of the pullback f*Γ onto the tensor product of KPK(Γ) and the group algebra of the kernel of the group completion of f. When Γ is strongly aperiodic, but need not be cofinal, every maximal tail T and every maximal ideal 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.
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BibTeX
@misc{nguyen2026maximal,
title = {{Maximal tails,character fibres and induced modules for pullback Kumjian-Pask algebras}},
author = {Bich Van Nguyen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20230},
eprint = {2609.20230},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20230},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Maximal tails,character fibres and induced modules for pullback Kumjian-Pask algebras
AU - Bich Van Nguyen
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.20230
UR - https://arxiv.org/abs/2609.20230
AB - Let f:N^k→N^ℓ be a surjective monoid homomorphism and let Γ be a row-finite ℓ-graph with no sources and finitely many vertices. We give an explicit graded isomorphism from the Kumjian--Pask algebra of the pullback f^*Γ onto the tensor product of KP_K(Γ) and the group algebra of the kernel of the group completion of f. When Γ is strongly aperiodic, but need not be cofinal, every maximal tail T and every maximal ideal 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.
ER -