The quantum algebra $U_q(s{l}_2)$ and its equitable presentation

Abstract

We show that the quantum algebra $U_q(s{l}_2)$ has a presentation with generators $x,x^{-1},y,z$ and relations $x{x}^{-1}=1$, $x^{-1}x=1$, $\frac{qxy-q^{-1}yx}{q-q^{-1}}=1$, $\frac{qyz-q^{-1}zy}{q-q^{-1}}=1$, $\frac{qzx-q^{-1}xz}{q-q^{-1}}=1$. We call this the equitable presentation. We investigate the action of $x,x^{-1},y,z$ on finite-dimensional $U_q(s{l}_2)$-modules.

Accepted for publication in J. Algebra