Equivariant Morita equivalence for finite cyclic group actions on noncommutative tori
Abstract
Li proved that the noncommutative tori Aθ and Agθ are strongly Morita equivalent whenever g∈SO(n,n|ℤ) and gθ is defined. We study equivariant extensions of these equivalences for canonical finite cyclic group actions. Let W∈GLn(ℤ) satisfy WtθW=θ, Wk=In and 1∉spec(W), and write ρ(R)=diag(R,(Rt)−1). If gρ(Wt)g−1=ρ(P) for some P∈GLn(ℤ), we equip Li's Agθ--Aθ Heisenberg equivalence bimodule with a compatible ℤk-action. Consequently, the crossed products AgθrtimesαPtℤk and AθrtimesαWℤk are strongly Morita equivalent. For the canonical flip action by ℤ2, we prove that Morita equivalence of crossed products implies Morita equivalence of the underlying tori in every dimension, and prove the converse in dimensions two through four. For each fixed dimension 2≤ n≤ 4, the complete invariants are the central rank and canonical K0-trace range of the underlying torus, with the latter taken up to positive scaling. For the standard cyclic group actions of orders $2$, $3$, $4$ and $6$ on rotation algebras, we prove that the crossed products are Morita equivalent exactly when the action orders agree and the parameters lie in the same fractional-linear GL2(ℤ)-orbit, generalizing the result of Bönicke et al.\ from irrational to arbitrary real parameters. We also construct infinite families of distinct parameters giving Morita-equivalent crossed products for each fixed W satisfying the stated hypotheses.
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BibTeX
@misc{chakraborty2026equivariant,
title = {{Equivariant Morita equivalence for finite cyclic group actions on noncommutative tori}},
author = {Sayan Chakraborty and Pratik Kumar Kundu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38695},
eprint = {2610.38695},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38695},
}
RIS
TY - UNPB
TI - Equivariant Morita equivalence for finite cyclic group actions on noncommutative tori
AU - Sayan Chakraborty
AU - Pratik Kumar Kundu
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38695
UR - https://arxiv.org/abs/2610.38695
AB - Li proved that the noncommutative tori A_θ and A_gθ are strongly Morita equivalent whenever g∈SO(n,n|ℤ) and gθ is defined. We study equivariant extensions of these equivalences for canonical finite cyclic group actions. Let W∈GL_n(ℤ) satisfy W^tθW=θ, W^k=I_n and 1∉spec(W), and write ρ(R)=diag(R,(R^t)^−1). If gρ(W^t)g^−1=ρ(P) for some P∈GL_n(ℤ), we equip Li's A_gθ--A_θ Heisenberg equivalence bimodule with a compatible ℤ_k-action. Consequently, the crossed products A_gθrtimes_α_P^tℤ_k and A_θrtimes_α_Wℤ_k are strongly Morita equivalent. For the canonical flip action by ℤ_2, we prove that Morita equivalence of crossed products implies Morita equivalence of the underlying tori in every dimension, and prove the converse in dimensions two through four. For each fixed dimension 2≤ n≤ 4, the complete invariants are the central rank and canonical K_0-trace range of the underlying torus, with the latter taken up to positive scaling. For the standard cyclic group actions of orders $2$, $3$, $4$ and $6$ on rotation algebras, we prove that the crossed products are Morita equivalent exactly when the action orders agree and the parameters lie in the same fractional-linear GL_2(ℤ)-orbit, generalizing the result of Bönicke et al.\ from irrational to arbitrary real parameters. We also construct infinite families of distinct parameters giving Morita-equivalent crossed products for each fixed W satisfying the stated hypotheses.
ER -