Year: 2014
Type: preprint
Abstract: The $n$-dimensional quantum torus $Λ$ is defined to be the $F$-algebra generated by variables $y_1, \cdots, y_n$ with the relations $y_iy_j = q_{ij}y_jy_i$ 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.
Source: arXiv (Cornell University)
Author: Gupta, Ashish
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W292988154