The equitable presentation for the quantum group $U_q(g)$ associated with a symmetrizable Kac-Moody algebra $g$

Abstract

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 $g=s{l}_2$ the equitable presentation has generators $X,X^{-1},Y,Z$ and relations $X{X}^{-1}=1$, $X^{-1}X=1$, $\frac{qXY-q^{-1}YX}{q-q^{-1}}=1$, $\frac{qYZ-q^{-1}ZY}{q-q^{-1}}=1$, $\frac{qZX-q^{-1}XZ}{q-q^{-1}}=1$.

16 pages. Submitted to J. Algebra