Quantizations of the extended affine Lie algebra $\widetilde{\frak{sl}_2(\mathbb{C}_q)}$

TL;DR

This paper presents three distinct quantizations of the extended affine Lie algebra sl_2(\u0101_q), resulting in new Hopf algebra structures, with some restrictions to related algebras.

Contribution

It introduces three novel quantizations of sl_2(\u0101_q), expanding the understanding of its algebraic structures and their interrelations.

Findings

Produced three noncommutative, noncocommutative Hopf algebra structures.

Established isomorphisms yielding three additional quantizations.

Restricted two quantizations to sl_2(\u0101_q).

Abstract

The extended affine Lie algebra $\widetilde{\frak{sl}_2(\mathbb{C}_q)}$ is quantized from three different points of view in this paper, which produces three noncommutative and noncocommutative Hopf algebra structures, and yield other three quantizations by an isomorphism of $\widetilde{\frak{sl}_2(\mathbb{C}_q)}$ correspondingly. Moreover, two of these quantizations can be restricted to the extended affine Lie algebra ${sl_2(\mathbb{C}_q)}$.