Classification of quasifinite representations with nonzero central charges for type A1 EALA with coordinates in quantum torus

In this paper, we first construct a Lie algebra L from rank 3 quantum torus, and show that it is isomorphic to the core of EALAs of type A1 with coordinates in rank 2 quantum torus. Then we construct two classes of irreducible Z-graded highest weight representations, and give the necessary and sufficient conditions for these representations to be quasifinite. Next, we prove that they exhaust all the generalized highest weight irreducible Z-graded quasifinite representations. As a consequence, we determine all the irreducible Z-graded quasifinite representations with nonzero central charges. Finally, we construct two classes of highest weight Z-graded quasifinite representations by using these Z-graded modules. Keyword: core of EALAs, graded representations, quasifinite representations, highest weight representations, quantum torus. §