Classification of quasifinite representations with nonzero central charges for type $A_1$ 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 $A_1$ with coordinates in rank 2 quantum torus. Then we construct two classes of irreducible ${\bf Z}$-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 ${\bf Z}$-graded quasifinite representations. As a consequence, we determine all the irreducible ${\bf Z}$-graded quasifinite representations with nonzero central charges. Finally, we construct two classes of highest weight ${\bf Z}^2$-graded quasifinite representations by using these ${\bf Z}$-graded modules.