Colloquium abstracts

Sachin Sharma
TIFR
September 5, 2013

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