The irreducible modules for the derivations of the rational quantum torus

Let $\mathbb{C}_q$ be the quantum torus associated with the $d
\times d$ matrix $q = (q_{ij})$,
$q_{ii} = 1$, $q_{ij}^{-1} = q_{ji}$, $q_{ij}$ are roots of unity, for all
$1 \leq i, j \leq d$ .
Let Der$(\mathbb{C}_q)$ be the Lie algebra of all the derivations of
$\mathbb{C}_q$.W.Lin and S.Lan defined a functor from $gl_d$-modules to
Der$(C_q)$modules.
They proved that for a finite dimensional irreducible $gl_d$-module $V$,
$V \otimes \mathbb{C}_q$ is a completely reducible Der$(C_q)$-module
except finitely many cases. In this talk we will show that $V \times
\mathbb{C}_q$ is an irreducible Der($\mathbb{C}_q) \ltimes
\mathbb{C}_q$-module which satisfies some conditions. The main aim of the
talk is to prove the converse of the above fact i.e., if $V'$ is an
irreducible $\mathbb{Z}^d$-graded Der($\mathbb{C}_q) \ltimes
\mathbb{C}_q$-module with finite dimensional weight spaces which
satisfies some conditions, then $V' \cong V \otimes C_q$ as
Der($\mathbb{C}_q) \ltimes \mathbb{C}_q$-module and when restricted to
Der($\mathbb{C}_q$), it is isomorphic to the module defined by W.Lin and S.Lan.

This is a joint work with Eswara Rao and Punita Batra.