Representations of the Dynamical Affine Quantum Group $U_{q,x}(\widehat{sl_2})=U_{q,\lambda}(\widehat{sl_2})$ and Hypergeometric Functions

TL;DR

This paper explores the representation theory of a specific dynamical quantum group, showing that intertwiners relate to well-poised hypergeometric functions, confirming a conjecture linking elliptic and basic hypergeometric series.

Contribution

It initiates the representation theory of the Hopf algebroid $U_{q,x}(\\widehat{sl_2})$ and proves the connection between intertwiners and $_{10}W_9$ hypergeometric functions, confirming a conjecture of Konno.

Findings

Intertwiner between tensor products is a well-poised $_{10}W_9$ symbol.

Degeneration of elliptic $_{12}V_{11}$ series to $_{10}W_9$ is established.

Representation theory confirms the degeneration as $p \to 0$.

Abstract

The representation theory of the Hopf algebroid $U_{q,x}(\widehat{sl_2})=U_{q,\lambda}(\widehat{sl_2})$ is initiated and it is established that the intertwiner between the tensor products of dynamical evaluation modules is a well-poised balanced $_{10}W_9$ symbol, confirming a conjecture of Konno, that the degeneration of the elliptic $_{12}V_{11}$ series to $_{10}W_9$ can be proven based on the representation theory of $U_{q,x}(\widehat{sl_2})$, the degeneration of the elliptic algebra $U_{q,p}(\widehat{sl_2})$ as $p \to 0$.