Elliptic Quantum Group Uq,p(ŝl2) and Vertex Operators

Introducing an H-Hopf algebroid structure into Uq,p(ŝl2), we investigate the vertex operators of the elliptic quantum group Uq,p(ŝl2) defined as intertwining operators of infinite dimensional Uq,p(ŝl2)-modules. We show that the vertex operators coincide with the previous results obtained indirectly by using the quasi-Hopf algebra Bq,λ(ŝl2). This shows a consistency of our H-Hopf algebroid structure even in the case with non-zero central element. 1 The Elliptic Algebra Uq,p(ŝl2) In this section we review a definition of the elliptic algebra Uq,p(ŝl2) and its RLL formulation following [1, 2]. 1.1 Definition of Uq,p(ŝl2) The elliptic algebra Uq,p(ŝl2) was introduced in [1] as an elliptic analogue of the quantum affine algebra Uq(ŝl2) in the Drinfeld realization. It was soon realized that Uq,p(ŝl2) is isomorphic to the the tensor product of Uq(ŝl2) and a Heisenberg algebra {P, e } [2]. We here define Uq,p(ŝl2) along the latter observation. Let us fix a complex number q such that q 6= 0, |q| < 1. Definition 1.1. [3] For a field K, the quantum affine algebra K[Uq(ŝl2)] in the Drinfeld realization is an associative algebra over K generated by the Drinfeld generators an (n ∈ 1 Z6=0), x ± n (n ∈ Z), h, c, d. The defining relations are given as follows. c : central , [h, d] = 0, [d, an] = nan, [d, x ± n ] = nx ± n , [h, an] = 0, [h, x (z)] = ±2x(z), [an, am] = [2n]q[cn]q n qδn+m,0, [an, x (z)] = [2n]q n qzx(z), [an, x (z)] = − [2n]q n zx(z), (z − qw)x(z)x(w) = (qz − w)x(w)x(z), [x(z), x(w)] = 1 q − q−1 ( δ ( q z w ) ψ(qw)− δ ( q z w ) φ(qw) ) . where [n]q = q−q q−q−1 , δ(z) = ∑ n∈Z z n and x(z) = ∑ n∈Z xn z , ψ(qz) = q exp ( (q − q) ∑ n>0 anz −n ) , φ(qz) = q exp ( −(q − q) ∑