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

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Abstract

For the $n$-dimensional multi-parameter quantum torus algebra $\Lambda_{\mathfrak q}$ over a field $k$ defined by a multiplicativelyantisymmetric matrix $\mathfrak q = (q_{ij})$ we show that, in the case whenthe 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 simplemodules. 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 withoutassuming the simplicity, and it is shown that a dichotomy still holds for theGK dimension of simple modules.

About the authors

Ashish Gupta

Ramakrishna Mission Vivekananda Educational and Research Institute

Email: a0gupt@gmail.com

Umamaheswaran Arunachalam

Harish-Chandra Research Institute

Email: ruthreswaran@gmail.com

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Copyright (c) 2022 Gupta A., Arunachalam U.

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