Representations of the Lie algebra of derivations for quantum torus (Q1881119)

Let \(\mathbb{C}_q\) denote the quantum torus associated to the \(d\times d\) matrix \(q=(q_{ij})_{1\leq i,j\leq d}\) with non-zero complex entries satisfying \(q_{ii} =1\) and \(q_{ij}^{-1}=q_{ji}\) for all \(1\leq i,j\leq d\). The authors define a functor from the category of \(\mathfrak{gl}_d(\mathbb{C})\)-modules to the category of \(\mathrm{Der}(\mathbb{C}_q)\)-modules which is a special case of the mixed product construction of \textit{Guangyu Shen} [Sci. Sin., Ser. A 29, 570--581 (1986; Zbl 0601.17013)]. The image of this functor consists of weight modules and the dimension of each weight space is the dimension of the correponding \(\mathfrak{gl}_d (\mathbb{C})\)-module. In the special case that \(q\) is the identity matrix this functor was introduced by \textit{T.A. Larsson} [Int. J. Mod. Phys. A 7, No. 26, 6493--6508 (1992; Zbl 0972.17502)]. The authors describe the structure of the image of a finite-dimensional irreducible \(\mathfrak{gl}_d(\mathbb{C})\)-module under this functor in the case that all the entries of \(q\) are roots of unity. In particular, the image of a finite-dimensional irreducible \(\mathfrak{gl}_d(\mathbb{C})\)-module \(V\) under this functor is completely reducible unless the highest weight of \(V\) is a fundamental weight or \(V\) is one-dimensional. In the case that all entries of \(q\) are \(1\), these results were already obtained by \textit{S. Eswara Rao} [J. Algebra 182, No. 2, 401--421 (1996; Zbl 0902.17012)].