Elliptic Quantum Group Uq,p(ŝl2), Hopf Algebroid Structure and Elliptic Hypergeometric Series

We propose a new realization of the elliptic quantum group equipped with the H-Hopf algebroid structure on the basis of the elliptic algebra Uq,p(ŝl2). The algebra Uq,p(ŝl2) has a constructive definition in terms of the Drinfeld generators of the quantum affine algebra Uq(ŝl2) and a Heisenberg algebra. This yields a systematic construction of both finite and infinite-dimensional dynamical representations and their parallel structures to Uq(ŝl2). In particular we give a classification theorem of the finite-dimensional irreducible pseudo-highest weight representations stated in terms of an elliptic analogue of the Drinfeld polynomials. We also investigate a structure of the tensor product of two evaluation representations and derive an elliptic analogue of the Clebsch-Gordan coefficients. We show that it is expressed by using the very-well-poised balanced elliptic hypergeometric series 12V11.