The positive even subalgebra of $U_q(\mathfrak{sl}_2)$ and its finite-dimensional irreducible modules

TL;DR

This paper introduces a presentation for a specific subalgebra of the quantum group $U_q(sl_2)$ and classifies its finite-dimensional irreducible modules when $q$ is not a root of unity.

Contribution

It provides a new presentation of the positive even subalgebra of $U_q(sl_2)$ and classifies its finite-dimensional irreducible modules.

Findings

Presented generators and relations for the subalgebra $\\mathcal{A}$.

Classified finite-dimensional irreducible modules of $\mathcal{A}$.

Applicable when $q$ is not a root of unity.

Abstract

The equitable presentation of $U_q(\mathfrak{sl}_2)$ was introduced in 2006 by Ito, Terwilliger, and Weng. This presentation involves some generators $x, y, y^{-1}, z$. It is known that $\{x^r y^s z^t : r, t \in \mathbb{N}, s \in \mathbb{Z}\}$ is a basis for the $\mathbb{K}$-vector space $U_q(\mathfrak{sl}_2)$. In 2013, Bockting-Conrad and Terwilliger introduced a subalgebra $\mathcal{A}$ of $U_q(\mathfrak{sl}_2)$ spanned by the elements $\{x^r y^s z^t : r, s, t \in \mathbb{N}, r+s+t \ {\rm even}\}$. We give a presentation of $\mathcal{A}$ by generators and relations. We also classify up to isomorphism the finite-dimensional irreducible $\mathcal{A}$-modules, under the assumption that $q$ is not a root of unity.