PRINCIPAL REALIZATION FOR THE EXTENDED AFFINE LIE ALGEBRA OF TYPE sl2 WITH COORDINATES IN A SIMPLE QUANTUM TORUS WITH TWO GENERATORS

We construct an irreducible representation for the extended affine algebra of type sl2 with coordinates in a quantum torus. We explicitly give formulas using vertex operators similar to those found in the theory of the infinite rank affine algebra A∞. Introduction. The purpose of this paper is to construct a module for the Extended Affine Lie algebra (EALA for short), we call it L̂, of type sl2 with coordinates in a quantum torus Cq. EALA’s were first introduced in the paper [HKT] as a natural generalization of affine Kac-Moody algebras and we note that the reported motivation for this work was from quantum gauge theories. Roughly one can think of EALA’s as higher dimensional generalizations of loop algebras, and in [HKT] all examples came from certain central extensions of the Lie algebra of polynomial maps of an n-dimensional torus T into a finite dimensional simple Lie algebra ġ over the field of complex numbers. That is, they used Laurent polynomials in many variables as their coordinates. Subsequently, in [BGK], it became clear that the set of axioms for EALA’s allowed other coordinate algebras as well. Indeed, the quantum torus Cq studied in [M], as well as certain alternative and Jordan algebra coordinates can appear as the coordinate algebra for an EALA. In this paper we focus on what is, in many respects, the easiest example of a coordinate algebra. Thus, our coordinates will be a quantum torus, Cq, parametrized by one nonzero scalar q which, in addition,we assume to be generic, in the sense that it is not a root of unity, and where Cq is generated by two elements s and t together with their inverses. These generators then satisfy ts = qst. The authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada Typeset by AMS-TEX 1 Such a quantum torus is simple as an algebra, and we note it is an algebraic version of the noncommutative torus studied in [C]. The module we construct is obtained through a process which follows the usual construction, by vertex operators, of the principal module for the affine Kac Moody Lie algebra A (1) 1 as found in [LW], [FLM], [K]. Thus, we identify a Heisenberg subalgebra Ĥ and use the standard irreducible module for this where the center acts as the identity to construct our module for L̂. Letting S = C[x1, x2, x3, . . . ] denote this Heisenberg module we will let