The quantum algebra $U_q(sl_2)$ and its equitable presentation
  We show that the quantum algebra $U_q(sl_2)$ 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 $U_q(sl_2)$-modules. 
Tatsuro Ito, Paul Terwilliger and Chih-wen Weng