VERTEX OPERATOR REPRESENTATIONS OF QUANTUM TORI AT ROOTS OF UNITY

Abstract

As Lie algebra, we add the center c1 (and the outer derivation d1) to the quantum torus [Formula: see text] to give the extended torus Lie algebra [Formula: see text] (and [Formula: see text] respectively). Before the present paper, only some level 1 vertex operator representations for some [Formula: see text] (and [Formula: see text]) were constructed. In this paper, we first give vertex operator representations for [Formula: see text] where I is an arbitrary index set. By embedding some [Formula: see text] into [Formula: see text], we obtain a series of higher level vertex operator representations for [Formula: see text] and [Formula: see text]. Most of these vertex operator representations yield irreducible highest weight modules over these [Formula: see text]. Also their character formulas follow directly.