A note on the $O_q(\hat{sl_2})$ algebra

TL;DR

This paper establishes a homomorphism linking the $O_q(\\hat{sl_2})$ algebra with the $q$-Onsager algebra, enabling new insights into quantum integrable models and special functions.

Contribution

It introduces an explicit homomorphism connecting the $O_q(\hat{sl_2})$ algebra to the $q$-Onsager algebra, facilitating applications in integrable models and special functions.

Findings

Derived the infinite $q$-deformed Dolan-Grady hierarchy.

Proposed higher Askey-Wilson relations for symmetric special functions.

Linked algebraic structures to quantum integrability and orthogonal polynomials.

Abstract

An explicit homomorphism that relates the elements of the infinite dimensional non-Abelian algebra generating $O_q(\hat{sl_2})$ currents and the standard generators of the $q-$Onsager algebra is proposed. Two straightforward applications of the result are then considered: First, for the class of quantum integrable models which integrability condition originates in the $q-$Onsager spectrum generating algebra, the infinite $q-$deformed Dolan-Grady hierarchy is derived - bypassing the transfer matrix formalism. Secondly, higher Askey-Wilson relations that arise in the study of symmetric special functions generalizing the Askey-Wilson $q-$orthogonal polynomials are proposed.