Highest weight irreducible representations of rank $2$ quantum tori

For any nonzero $q\in C $ (the complex numbers), the rank $2$ quantum torus $ C _q$ is the skew Laurent polynomial algebra $ C [t_1^{\pm1}, t_2^{\pm1}]$ with defining relations: $t_2t_1=qt_1t_2$ and $t_it_i^{-1}=t_i^{-1}t_i=1$. Here we consider $ C _q$ as the naturally associated Lie algebra. We add the one dimensional center $ C c_1$ and the outer derivation $d_1$ to $ C _q$ to get the extended torus Lie algebra $\widetilde{ C }_q$ (and $\widehat{ C }_q$, in a different manner), where we assume $q$ is a primitive $m$-th root of unity for $\widehat{ C }_q$. Before this paper, there appeared highest weight representations for $\widetilde{ C }_q$ and $\widehat{ C }_q$ with only positive integral levels. In this paper, we define the highest weight irreducible ($\hbox{$Z\hskip -5.2pt Z$}$-graded) module $V(\phi)$ over $\widetilde{ C }_q$ and $\widehat{ C }_q$ for any linear map $\phi: C [t_2^{\pm1}]+ C c_1+ C d_1\to C $ thus the central charge (level) can be any complex numbers. We obtain the necessary and sufficient conditions for $V(\phi)$ to have finite dimensional weight spaces, thus obtaining a lot of new irreducible weight representations for these Lie algebras. The corresponding irreducible $\hbox{$Z\hskip -5.2pt Z$}\times\Z$-graded modules with finite dimensional weight spaces over $\widetilde{ C }_q$ are also constructed.

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Published 1 January 2004