The quantum algebra $U_q(sl_2)$ and its equitable presentation
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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.
Accepted for publication in J. Algebra
Accepted for publication in J. Algebra