Elliptic quantum group Uq,p(sl̂2), Hopf algebroid structure and elliptic hypergeometric series

We propose a new realization of the elliptic quantum group equipped with the HH-Hopf algebroid structure on the basis of the elliptic algebra Uq,p(sl̂2). The algebra Uq,p(sl̂2) has a constructive definition in terms of the Drinfeld generators of the quantum affine algebra Uq(sl̂2) and a Heisenberg algebra. This yields a systematic construction of both finite- and infinite-dimensional dynamical representations and their parallel structures to Uq(sl̂2). 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 V1112.