Finite groups acting on higher dimensional noncommutative tori

2014/02/08 by Ja A Jeong, Jeong, Ja A, Jae Hyup Lee +1 · 1 citation
Mathematics · Physics and Astronomy · #22D25 #46L35 #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1402.1826

openalex publication_date 2014/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For the canonical action α of SL₂(ℤ) on 2-dimensional simple rotation algebras A_θ, it is known that if F is a finite subgroup of SL₂(ℤ), the crossed products A_θ\rtimes_αF are all AF algebras. In this paper we show that this is not the case for higher dimensional noncommutative tori. More precisely, we show that for each n≥ 3 there exist noncommutative simple ϕ(n)-dimensional tori A_Θ which admit canonical action of ℤ_n and for each odd n≥ 7 with 2ϕ(n)≥ n+5 their crossed products A_Θ\rtimes_αℤ_n are not AF (with nonzero K₁-groups). It is also shown that the only possible canonical action by a finite group on a 3-dimensional simple torus is the flip action by ℤ₂. Besides, we discuss the canonical actions by finite groups ℤ₅, ℤ₈, ℤ₁₀, and ℤ₁₂ on the 4-dimensional torus of the form A_θ⊗ A_θ.

Citations

• Projective Modules over Higher-Dimensional Non-Commutative Tori

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