Elliptic Quantum Group U_{q,p}(\hat{sl}_2), Hopf Algebroid Structure and Elliptic Hypergeometric Series

TL;DR

This paper introduces a new realization of the elliptic quantum group U_{q,p}(\\hat{sl}_2) with a Hopf algebroid structure, providing systematic constructions of representations and connecting tensor products to elliptic hypergeometric series.

Contribution

It offers a systematic construction of the elliptic quantum group U_{q,p}(\hat{sl}_2), including classification of finite-dimensional representations and elliptic analogues of Clebsch-Gordan coefficients.

Findings

Classification of finite-dimensional irreducible pseudo-highest weight representations.

Expression of elliptic Clebsch-Gordan coefficients via elliptic hypergeometric series.

Construction of dynamical representations paralleling U_q(\ar{sl}_2).

Abstract

We propose a new realization of the elliptic quantum group equipped with the H-Hopf algebroid structure on the basis of the elliptic algebra U_{q,p}(\hat{sl}_2). The algebra U_{q,p}(\hat{sl}_2) has a constructive definition in terms of the Drinfeld generators of the quantum affine algebra U_q(\hat{sl}_2) and a Heisenberg algebra. This yields a systematic construction of both finite and infinite-dimensional dynamical representations and their parallel structures to U_q(\hat{sl}_2). 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.