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

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 $\{x^r{y}^s{z}^t:r,s,t\in \mathbb{N},r+s+t\mathrm{even}\}$. 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.

22 pages

References in corpus (1)

• The Universal Askey-Wilson Algebra and the Equitable Presentation of $U_q({\mathfrak{sl}}_2)$