Highest weight irreducible representations of rank $2$ quantum tori

Abstract

For any nonzero q ∈ C (the complex numbers), the rank 2 quan- tum torus Cq is the skew Laurent polynomial algebra C(t ±1 1 ,t ±1 2 ) with defining relations: t2t1 = qt1t2 and tit −1 i = t −1 i ti = 1. Here we consider Cq as the naturally associated Lie algebra. We add the one dimensional center Cc1 and the outer derivation d1 to Cq to get the extended torus Lie algebra � Cq (and � Cq, in a different manner), where we assume q is a primitive m-th root of unity for � Cq. Before this paper, there appeared highest weight representations for � Cq and � Cq with only positive integral levels. In this paper, we define the highest weight irreducible (Z Z-graded) module V (φ) over � Cq and � Cq for any linear map φ : C(t ±1 2 )+ Cc1 + Cd1 → C, thus the central charge (level) can be any complex numbers. We obtain the necessary and sufficient conditions for V (φ) to have finite dimensional weight spaces, thus obtaining a lot of new irreducible weight repre- sentations for these Lie algebras. The corresponding irreducible Z Z × Z Z-graded modules with finite dimensional weight spaces over � Cq are also constructed.