A note on modules over the quantum torus

2014/12/31 by Ashish Gupta, Gupta, Ashish
Mathematics · Physics and Astronomy · #16T20 #17B37 #20G42 #20G42 16T20 #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum many-body systems #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1501.00072

openalex publication_date 2014/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The n-dimensional quantum torus Λ is defined to be the F-algebra generated by variables y₁, ⋯, y_n with the relations y_iy_j = q_ijy_jy_i where q_ij are suitable scalars from the base field. This algebra is also the twisted group algebra of the free abelian group A on n generators. Each subgroup of corresponds to a sub-algebra of the quanutm torus. A may contain non-trivial subgroups B so that the corresponding sub-algebra is commutative. In this paper we show that whenever the quantum torus Λ has center F, a Λ module M that is finitely generated over such a commutative sub-algebra U is necessarily torsion-free over U and has finite length. We also show that M has finite length. We also apply tbis result to modules over infinite nilpotent groups of class 2.

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