The equitable presentation for the quantum group $U_q(g)$ associated   with a symmetrizable Kac-Moody algebra $g$
  We consider the quantum group $U_q(g)$ associated with a symmetrizable Kac-Moody algebra $g$. We display a presentation for $U_q(g)$ that we find attractive; we call this the equitable presentation. For $g=sl_2$ the equitable presentation has generators $X,X^{-1},Y,Z$ and relations   $X X^{-1} =1$,   $X^{-1} X=1$,   $\frac{qXY-q^{-1}YX}{q-q^{-1}}=1$,   $\frac{qYZ-q^{-1}ZY}{q-q^{-1}}=1$,   $\frac{qZX-q^{-1}XZ}{q-q^{-1}}=1$. 
Paul Terwilliger