By looking at defining relations of standart deformation of $sl_2$, which is:

$[E,F] = \frac{q^{H}-q^{-H}}{q-q^{-1}}$, $[H,E] = 2E$ and $[H,F] = -2F$,

some questions come around.

For example, one can check that Jakobi identity is satisfited, but it would be also satisfited if one considers arbitary function $Fun(D)$ instead of original one $\frac{q^{H}-q^{-H}}{q-q^{-1}}$.

I know, that for the algebra to be quantum, the conditions of Hopf algebra have to be enjoyed.

But still, can we imaging another deformation of $sl_2$? Or there is a theorem. which says that it is the only deformation?

Thanks in advance!