By looking at defining relations of standard deformation of $\mathfrak{sl}_2$, which are:

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

some questions come around.

For example, one can check that Jacobi identity is satisfied, but it would be also satisfied if one considers an arbitrary function $Fun(D)$ instead of the 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 imagine another deformation for $\mathfrak{sl}_2$? Or there is a theorem which says that it is the only deformation?

Thanks in advance!