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

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.