The algebra $U_q({\mathfrak{sl}_2})$ in disguise

TL;DR

This paper explores the relationship between the quantum algebra $U_q(sl_2)$ and $q$-Racah type tridiagonal pairs, revealing new presentations of a subalgebra through module constructions.

Contribution

It introduces two novel nonstandard presentations of the subalgebra $U_q^igvee$ using tridiagonal pairs of $q$-Racah type and analyzes their properties.

Findings

Constructed two finite-dimensional modules of $U_q^igvee$

Derived two nonstandard presentations of $U_q^igvee$

Analyzed the structure and relations of these presentations

Abstract

We discuss a connection between the algebra $U_q({\mathfrak{sl}_2})$ and the tridiagonal pairs of $q$-Racah type. To describe the connection, let $x,y^{\pm 1},z$ denote the equitable generators for $U_q({\mathfrak{sl}_2})$. Let $U^\vee_q$ denote the subalgebra of $U_q({\mathfrak{sl}_2})$ generated by $x,y^{-1},z$. Using a tridiagonal pair of $q$-Racah type we construct two finite-dimensional $U^\vee_q$-modules. The constructions yield two nonstandard presentations of $U^\vee_q$ by generators and relations. These presentations are investigated in detail.