Representations of the $n$ dimensional quantum torus

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 $x_i{x}_j=q_{ij}{x}_j{x}_i$, where $\mathbf{q}=(q_{ij})$. We show that the modules that are finitely generated over certain commutative sub-algebras $\mathcal{B}$ are $\mathcal{B}$-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.

10 pages