Representations of the $n$ dimensional quantum torus

TL;DR

This paper studies the structure and dimensions of modules over the n-dimensional quantum torus algebra, revealing conditions for torsion-freeness, finite length, and Gelfand-Kirillov dimensions of simple modules.

Contribution

It characterizes finitely generated modules over certain sub-algebras, determines Gelfand-Kirillov dimensions of simple modules, and establishes existence results for modules satisfying specific dimension inequalities.

Findings

Modules finitely generated over certain sub-algebras are torsion-free and have finite length.

Gelfand-Kirillov dimensions of simple modules are either 1 or satisfy a specific lower bound.

Existence of simple modules satisfying the Gelfand-Kirillov dimension inequality is proven.

Abstract

The $n$-dimensional quantum torus $\mathcal O_{\mathbf q}((F^\times)^n)$ is defined as the associative $F$-algebra generated by $x_1, \cdots, x_n$ together with their inverses satisfying the relations $x_ix_j = q_{ij}x_jx_i$, where $\mathbf q = (q_{ij})$. We show that the modules that are finitely generated over certain commutative sub-algebras $\mathscr B$ are $\mathscr B$-torsion-free and have finite length. We determine the Gelfand-Kirillov dimensions of simple modules in the case when \[ \Kdim(\mathcal O_{\mathbf q}((F^\times)^n)) = n - 1, \] where $\Kdim$ stands for the Krull dimension. In this case if $M$ is a simple $\mathcal O_{\mathbf q}((F^\times)^n)$-module then $ \gk(M) = 1$ or \[ \gk(M) \ge \gk(\mathcal O_{\mathbf q}((F^\times)^n)) - \gk(\mathcal Z(\mathcal O_{\mathbf q}((F^\times)^n))) - 1,\] where $\mathcal Z(C)$ stands for the center of an algebra $C$. We also show that there always exists a simple $F \s A$-module satisfying the above inequality.