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October 26, 2016
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Kaiming Zhao

Highest weight irreducible representations of rank 2 quantum tori, Mathematical Research Letters, Vol.11, No.5-6, 615-628(2004)

S. Eswara Rao, K. Zhao

published: 2004 | Research publication | Research Paper

For any nonzero $qin C$ (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 ($$-graded) module $V(phi)$ over $widetilde{C}_q$ and $widehat{C}_q$ for any linear map $phi:C[t_2^{pm1}]+C c_1+C d_1 oC$ where 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 $ imes$-graded modules with finite dimensional
weight spaces over $widetilde{C}_q$ are also constructed.

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