Gelfand-Kirillov dimensions of simple modules over twisted group algebras $k \ast A$

Abstract

For the $n$-dimensional multi-parameter quantum torus algebra $\Lambda_{\mathfrak q}$ over a field $k$ defined by a multiplicatively antisymmetric matrix $\mathfrak q = (q_{ij})$ we show that, in the case when the torsion-free rank of the subgroup of $k^\times$ generated by the $q_{ij}$ is large enough, there is a characteristic set of values (possibly with gaps) from $0$ to $n$ that can occur as the Gelfand-Kirillov dimensions of simple modules. The special case when $\mathrm{K}.\dim(\Lambda_{\mathfrak q}) = n - 1$ and $\Lambda_{\mathfrak q}$ is simple, studied in A. Gupta, $\mathrm{GK}$-dimensions of simple modules over $K[X^{\pm 1}, \sigma]$, Comm. Algebra, 41(7) (2013), 2593-2597, is considered without assuming the simplicity, and it is shown that a dichotomy still holds for the GK dimension of simple modules.