On the second realization for the positive part of $U_q(\widehat{sl_2})$ of equitable type

TL;DR

This paper introduces and analyzes a second realization of the positive part of the equitable presentation of $U_q(\,widehat{sl_2})$, providing explicit algebraic structures, homomorphisms, and relations to Drinfeld's realizations.

Contribution

It presents a new second realization of the positive part of the equitable quantum algebra $U_q(\,widehat{sl_2})$, including explicit presentations and homomorphisms.

Findings

Explicit presentation of $U_q^{T,+}$ using a K-operator

Construction of an injective homomorphism to Drinfeld's realization

Characterization of the comodule algebra structure and central extension

Abstract

The equitable presentation of the quantum algebra $U_q(\widehat{sl_2})$ is considered. This presentation was originally introduced by T. Ito and P. Terwilliger. In this paper, following Terwilliger's recent works the (nonstandard) positive part of $U_q(\widehat{sl_2})$ of equitable type $U_q^{IT,+}$ and its second realization (current algebra) $U_q^{T,+}$ are introduced and studied. A presentation for $U_q^{T,+}$ is given in terms of a K-operator satisfying a Freidel-Maillet type equation and a condition on its quantum determinant. Realizations of the K-operator in terms of Ding-Frenkel L-operators are considered, from which an explicit injective homomorphism from $U_q^{T,+}$ to a subalgebra of Drinfeld's second realization (current algebra) of $U_q(\widehat{sl_2})$ is derived, and the comodule algebra structure of $U_q^{T,+}$ is characterized. The central extension of $U_q^{T,+}$ and its relation with Drinfeld's second realization of $U_q(\widehat{gl_2})$ is also described using the framework of Freidel-Maillet algebras.