Affiliation:
1. Department of Mathematics, Ramakrishna Mission Vivekananda Educational and Research Institute (RKMVERI), India
2. Harish-Chandra Research Institute, India
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.
Funder
Harish-Chandra Research Institute
NBHM
Publisher
Steklov Mathematical Institute
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